I spend a great deal of my time in these places (and have done for many years).
Each company handles the situation differently but the salient point is this: It is possible (and common) for any air carrier, logistics company, consulting company, or whatever to get insurance coverage for employees operating in war zones. Yes of course it costs more, but it is commonly done. (Otherwise many of us would not go!)
Having said that, I have experienced the situation where a company stated they had the coverage and it turned out they did not. When starting out with a new company, or making your first trip into one of these places, ALWAYS ask to see, and make a copy of, the actual insurance coverage document. If they have it, the human resources staff are quite used to fulfilling those requests.
Lastly, don't assume this applies only to the more well known conflict areas (Afghanistan, Iraq, certain areas of Pakistan, etc). There are many areas in Africa that are considered by insurers to be "conflict zones."
FWIW . . .
grizz