I think even if you can't get away from the show and bus too much you've got a good opportunity to build a library of genuinely useful recordings.
If you can take your recording kit into the venues you'd probably have an opportunity not many field recordists get, to record the sound of music venues without people or music! That should yield some interesting recordings and even if there are people milling around you'll get some good ambiences. You might also get some nice stuff of setup / breakdown. I imagine you'd get decent metal impacts from scaffolding / stages being put up and taken apart. The backstage areas might also yield some interesting room tones.
Then there are the general ambiences you'll get. Imagine how many restaurants, diners, cafes, pubs and bars you'll go into. All the busy main streets and quiet back streets you'll come across. If you're staying in any high rise hotels (that let you open the windows!) you'll get some great skylines. If you're not too busy setting up before the gigs there'll be all the crowds outside the venues.
Basically if you can get a portable recorder and keep it on you all the time you should get tons of great stuff. Make sure you buy enough SD cards and an external hard drive to back up on to as you go!
The other thing is, if you know your route around the country already, you could have a look along the route using Google maps to see if there is anything unusual or unique near where you're going. Maybe there's a cave system near one of your stops that you could go into and record some ambiences in on a day off. Maybe there's a clock museum that you could go into and record some mechanisms. Maybe there's a big tower that you could go up and record some distant skylines and winds.
Definitely +1 to @christiancoriolis on recording "mundane" things like hotel room tones and specifics. Record all the doors and rooms you can! Room tones and doors are the sort of things that you can never have enough of. After 5 months you should have a pretty great library of room tones and doors!
Also +1 to @Arnoud Traa and @christiancoriolis for having a plan for the editing period of this project. You can record hours of stuff but if you never edit and organise it you might as well not bother. Verbal slating is invaluable for this, it means you never have to worry about losing your notebook or even making notes if you don't want to, you'll have all the information in the recording itself. It's useful to have a notebook too though, especially if you're doing stealth recordings in public places where you don't want to blow your cover by verbally slating a take!
As @christiancoriolis says the Sony PCM series all seem to be very popular with field recordists as a small high quality solution. I'm sure you could find a used one for a good price, there's always the older D50 as well. Just make sure you budget for a Rycote (or similar) wind jammer and possibly some kind of tripod to reduce handling noise (Manfrotto Nano stands are great). I think Rycote also make a pistol grip for portable recorders as well. Really only the wind jammer is essential though.
Good luck, I'm getting excited about the trip for you!