Here's what I would do: tell the second company (the one you expect an invite to fly out and interview) that you will already be in town for another interview. Ask them if they'd like to interview you in-person the day before or day after your other interview with the first company.
If they agree (and they should--it will save them money on a flight), ask the first company if they are flexible with travel dates. In most cases they will be (I have first-hand experience with this). Tell them you'd like to spend some time getting to know the city, and ask for them to push the flight date in order to accommodate your other interview.
The only downside is you may be on the hook for the additional night at the hotel. You may be able to get that money out of the second company. They already know you have the interview, so be forthcoming and ask them to pay for the extra night (they're already saving on the flight costs).
As an aside: one benefit of this method is it has the potential to push the second job to speed up their process in order to interview you faster. And then you'll have the opportunity to have both of the offers on the table at the same time.