Yes they should. Other details would be relevant, whether they're driving together, other coworkers going too, hotel, dinners? For example if my husband was going out of town with just another female and driving together.. 

In my case, yes only because we talk about everything.  So for DH to say he has to go out of town for work for a few days, I would be interested.  "Oh, interesting.  Where you headed?  Are you staying some place nice?  What kind of stuff do you have to do while you're there?  Will yo uat least have a chance to go out with friends for a nice dinner?"  That kind of stuff. And naturally it would come up in conversation  "Yup, staying at a Best Western.  Diane and I are hoping to try that new hot wings restaurant".  

Would he make a specific point of saying "I'm going to work out of town with Diane, but don't worry we have separate rooms and won't be spending any time together socially".  No.  And I'd be wierded out if he did.  He has female friends.  He's welcome to go out with them in public.  As a senior staff member he wouldn't take the risk of being anywhere private (not worth the potential false accusations).  But he has never given me a moment's worry so my answer may not be your answer.

In my marriage, yes DH and I would let each other know if a co-worker was travelling with us out of town and I would expect he disclose if it was a woman too. 
But DH is such a shit stirrer though, to wind me up, he wouldn't be able to control himself and immediately tell me how he would wine and dine her jokingly.
I trust him 100% and wouldn't be threatened at all only because he makes me feel loved and has made me feel secure. 
Example, Once he went into a cafe to order our  coffees and he was in there for ages! I was sitting with DD4 and DS1. Then a slim, blond attractive woman approached me with a baby strapped to her front and apologised for holding up my husband's time as she used to go to school with him and stated she was in there chatting to him for ages. I smiled and waved to her good bye when she left and when DH sat down he told me up front she told him that she lives local here and for him to let her know when he's here again and they can catch up. Lol who says that to an old high school friend when we are in our late 30's? I just shook my head and said of course she did, we laughed it off. But, funny she never mentioned to me she invited my husband to catch up. So, it goes back to trust and knowing your SO will disclose information to you. 

So it sounds like you specifically asked if this woman is going out of town ever and he lied and said no however you have proof she is asking what hotel he is going to.. This doesn't look good. It could be innocent, he doesn't want to upset you... However at minimum he's lying about a woman.

I would check their text thread again. I know others may not agree however she already knows he's telling lies about his behavior

I would take a photo of the text where she's asking him what hotel he's staying at. Then as normal again ask if a specific male colleague is going, followed by the woman's actual name and ask directly if she's going on this trip. If he lies again I guess you have a decision to make whether to confront about the lie. 

Projecting my own experience here, but I ignored gut feelings and lies (that I KNEW were lies) about another woman and rationalized it exactly the same way you are, that he didn't want to upset me. When I was finally prepared and strong enough to really dig, I was horrified by how far into an affair they were, and the lengths he had gone to in order to deceive me. The betrayal was bad enough, but the lies on top of lies were really what did my head in. It's taken years to recover from the trauma of being lied to so much, and even still I'll occasionally be reminded of a lie and it will trigger me. 

All that being said, I would always recommend that you trust your intuition. And that you have your mind made up what you're going to do if/when you confront him. 

Yeah ideally they should but some men don't like confrontation and they think "oversharing" could lead them down the path of multiple arguments and no p*ssy 

So they "lie" by omission to spare your feelings (if they assume you can't handle the truth) so you remain in ignorant bliss

He might not be planning to have relations with another woman but the way he's going about it with regards to the lack of details he's providing you is not ideal. 

I would look at the phone bill and see for exactly long they've been speaking on the phone. You should also match up the text messages that appear on the phone bill and his text messages. If it doesn't match and he's deleting texts, there's obvously more to the story. A person who cheats or is thinking of cheating cannot be expected to be forthcoming and honest so unfortunately the onus is on you to dig. 
The fact that he lied that she was going is not good, but I know you know that. If I were you I would be on high alert about this. 
My honest thoughts about this are that maybe initially it was nothing, but maybe on company trip 1, they saw each other and one, the other or both had an attraction. This likely led to flirtatious behavior, maybe hanging out past work, and looking forward to work trip number 2. Not sure how far those 2 went, but from the way you described it, I don't think those early morning talks are about work. 

One thing I will say is at work there's no such thing as too early if it's work related and they start early. I started work at 6am and send any messages I need to. Having said that I just don't like how he treated you with the lying and it feels a bit off. Also flirting is different then just work  questions. I don't like that she texted asking him what hotel.

I guess it feels different on Microsoft teams Vs a text to a phone? One woman texted my husband at 5 am when he was filling in as boss and just said, "Good morning this is Ashleee" and I was like who the heck is Ashlee and why she be texting at 5 am

I would say keep an eye out. His lying is certainly NOT ok no matter what the reason. Even if he thinks you'll be upset. He still has to know to communicate.

So yes not going out of town is the answer but it isn't logical. 

Sorry, but not going out of town is NOT the answer.The problem is not who is going on these business trips, the problem is him and his lying.There is a reason he lied and that is the issue. As well as the mistrust between the two of you.

If there is truly nothing going on, then he lied because there is a big lack of trust between the two of you.You don't trust him, and he doesn't trust you, that you would believe that nothing is going on with his co-worker. It's not about protecting you, he lied to protect himself, to avoid any conflict. But, why would there have been any conflict? Why would he think that you would overreact? Co-workers of the opposite sex go out of town on business all of the time. If you trusted him, and he trusted that you trusted him, why would this be something to lie about? Why would he not just stand his ground that nothing is going on? Unless the question is, "Does my ass look fat?", lying is not the answer.

I would be weary of being in a relationship with a man who has so little resolve and self respect. His weakness when it comes to BM and the kids, may carry over into other aspects of his life, and other people. He needs to man up.

And if you don't trust him, why are you still in this relationship, with all of the other issues? You deserve a relationship without so much stress, worry, and mistrust. Relationships aren't always easy but they shouldn't be this hard, either.

I travel for business several times a year.  I work in a male dominated field where most supplier visits and conferences will be 80% men.  My department is all men, except me and an admin (who doesn't travel.. no need).

When we go, we always stay at the same hotel (separate rooms.. haha).  We do it because it is either the conference hotel so it has a special rate.. or logistically, we are traveling together.. so can just share one rental car.. share ubers.. coordinate with the supplier for one ride to the site etc.. 

I will say, that I don't think I have ever specifically made a point of telling my DH I'm "traveling with men".. but the assumption is just there I guess.. and sometimes I mention which of my coworker men is going to be on the trip.. or if I'm going alone (occasionally).  So, my DH knows and has never shown any concern.. and sometimes I only can contact him once or twice a day.. get in after late nights of dinner and drinks with the clients/coworkers.. he knows that's part of the program.

So, I guess the mistrust can be there for some.. but for my DH and myself.. it has not been an issue.. and he has worked offshore where there were women on the boat too.. again.. no reason for either of us to be concerned.

Another reason for her asking for "what hotel".. is if she is tasked with doing any coordination for the trip.. planning dinner with clients.. arranging car rental near the hotel.. or it could be that she is familiar with the place he is traveling with.. and is just asking out of curiosity.. to give him a recommendation of a good restaurant near the hotel. 

So, those are all the kinds of reasons why you  might have seen that message.. but it also feels like you have some general mistrust with him.. and given the state of things with your fighting over his kid's sports.. maybe it is more reasonable to assume he is seeking out attention outside your relationship.

In the end.. you could do a couple of things.

1.  Travel out there with him... or travel out to "surprise" him.

2.  Ask him directly who else is going on the trip.

EDIT... I went and read more of the posts. and saw he did admit it.. but is claiming that it's because you would overreact that he didn't tell you.

My personal feeling is that if THIS is what he thinks of you.. then the relationship is pretty broken.. 

Because.. your first instinct is that her going on the trip and their contact is not "professional" in nature.. and his first instinct is to hide that she is going because he views you as someone who overreacts.. and he doesn't respect you enough to see you as mature enough to handle knowing that he might be traveling with a coworker who is a woman.

I mean.. at the base of it.. kind of ridiculous.. to hide that he is going with her.. just like when I travel.. or my DH travels.. nothing to stop us from cheating if we were so inclined.. with a totally unrelated person.. right?  

I don't think he respects your feelings.. or your perspectives.. he thinks you overreact about his kids.. he thinks you would overreact about this.. he doesn't trust you as a partner to have information.. and perhaps the conflicts about his kids have made him consider moving on at some point.. but he isn't ready to pull that trigger yet?

If you want to stay with him.. therapy.. and lots of it.

He keeps saying it is because of the way I would react. And he is saying that the way I am reacting now is the way I would have if he told me in the first place. 

Total cop out and not a valid reason for lying. He doesn't get to say that he knew how you would react. If you would have reacted that way, had he told the truth, then that is something that would have had to have been dealt with. Lying to avoid that is bullshit and doesn't do anything positive for the relationship.

The only way a relationship truly works is to be real. He needed to give you the chance to react however you would have reacted. He cannot second guess that reaction, and lie to avoid whatever it might be. That's not fair to you or the relationship. He can't expect you to feel secure in a relationship where he is lying because he wants to avoid how he thinks you will react. Where does it stop? How do you believe anything he says?

Also, I go back to the question of why did he think you would overreact? Why does he not have enough trust in you, that you would accept that it's all just business?

I understand that he's been cheated on but he cannot bring that baggage into this relationship. Not everybody is a cheater, not everybody is suspicious and accusing.

If he wants to be in an adult relationship, he needs to deal with whatever comes up, however you would react to whatever situation. He needs to be honest and real, and allow you to be honest and real. Because, truthfully, if two people can't be real with each other, they have no business being in the relationship.