Long but leisurely trip from NH to NM up to MT then home

Depending where you're planning to go in those areas, if you want to visit a national park, or will be somewhere over a major holiday, I strongly suggest reservations. Yes, people do find cancellations and other openings available, but unless you don't mind changing your plans and perhaps missing out on someplace you especially want to see, you should consider doing so. Sounds like a good trip, perhaps a bit rushed for a month. Safe travels!