I'm not private-sector, and the travel budget is "as little as possible." So, in general: cheapest or close-to-cheapest available fares, period.

I do have the option of buying a more expensive fare and being reimbursed the amount of the cheapest one.

Travel days count as a half-day for per diem purposes, and as a full day if the travel is more than 12 hours one-way. (This is common. My record is somewhere over 50 hours one-way.)

Layovers of greater than 6 hours rate a lounge pass.
I think I get a hotel room if the layover is greater than 12 hours.
I do get to keep my own miles.

The strategy for dealing with this scenario, which I have embraced fully and am promoting to my colleagues, is as follows:

Find an airline that consistently has competitive fares from where you are to a lot of other places. Join its FF program. Tell the flight coordinator if you can get better fares logged into its web site. Book your personal travel on it, too. Get elite status ASAP. Get as much elite status as you possibly can. And maintain it.

Doing crazy-long trips to places that I can't mention to my friends without them nervously changing the subject is a lot more survivable with pre-boarding, preferred seating, free upgrades, lounge access, and all that nice stuff. And when my colleagues find out that I flew in for an assignment up front, they want it too.