So, there have been a couple of questions about landing at places other than military airfields. Not a military guy, only a private pilot, the father of a military pilot, and an aviation buff, but here are my insights/experiences.
You've been given some good answers in previous posts. As mentioned, fighters generally have pretty short legs, so they aren't going very far. A 'tanker' drag is pretty common, and they pass over the Amarillo area pretty regularly. A flight of fighters will follow a tanker cross country, refueling as they go. MFP did this going to Red Flag at Nellis, and her squadron did it over the Atlantic on a TDY to Europe.
I've related the apocryphal story of the squadron on a tanker drag overseas, where the wives were riding in the passenger section of the KC-135. (Pre-female fighter pilots) Wives decided to give their hubbys a thrill, and as their man slotted up to refuel, they would go to the boomer station, and flash their boobs. Nothing said on the radio, just a nice surprise. Then they felt sorry for the lone bachelor in the squadron, and the CO's wife decided she would take one for the team, and she would flash him. Said pilot apparently took several tries to hook up to the boom, and really couldn't say on the radio what was distracting him.
Some airfields (Kirtland AFB/Albuquerque, Killen/Robert Grey AAF) are joint military/commercial. Lots of others (Amarillo) are former military bases with attractive facilities, such as long runways and/or good locations.
We get a LOT of military traffic at Amarillo, especially Army and Navy. The local aviation spotters love it. Long runway, halfway between the coasts, FBO with a lot of experience with military aircraft. I believe that proper fueling is important, but the availability of 'start carts' is a plus, so that aircraft don't have to depend on internal batteries to start.
Funny thing is that we don't get a lot of Air Force fighter traffic stopping, though we get lots of touch and goes, and instrument approaches from trainers, as we are close to training bases at Vance, Altus, and Sheppard AFBs. When MFP stopped by while ferrying an F-16 from Hill AFB to Shaw AFB, she got a lot of attention, as F-16s are rare at Amarillo. Female pilot also got people's curiosity up.
However, when she left, she had a pressurization failure. She was going to have to park the plane overnight, and could not leave it at Amarillo. She diverted to Wichita Falls (Sheppard AFB), parked it, and some poor Air Police guy had to guard it until maintenance could fly some folks over to fix it. I've seen trainers and even cargo planes overnight at Amarillo, but they don't leave fighters on civilian airfields without security, if at all possible.
Along those lines, when she did a FAIP tour, they generally did not do extremely long XC flights. I think the guidance was to land at an airport or base no more than 4-6 hours away by car, so you could return easily if the plane broke.
Finally, I've seen a few news releases where they have taken fighters and fuel trucks to remote airfields as part of a SHTF exercise. Basically teaching pilots how to help refuel and rearm their aircraft if their main base was disabled in wartime.
If you do the Facebook thing, here is the Amarillo Plane Spotter page. They catch some really unusual aircraft coming by, as well as the Ospreys and Cobra helicopter upgrade (stupidly called Viper) that are produced in Amarillo.
https://www.facebook.com/groups/128291860579641