I recently stayed at some super sweet cabins in Cody Wyoming - behind the Holiday Inn - Buffalo Bill cabins. State of the art cabins, walking distance to lots of stuff. Kind of expensive right now, but we took two cabins (two couples) for 91 each on expedia last month, while their official website was advertising 179 per night..
from Cody , chief joseph to Beartooth to Red Lodge is a must, in my opinion. The park is going to be full of people, and slow, but if you haven't seen it you probably have to see it.
If you do get into the park, a soak in the boiling river is pretty cool. (the actual location called "the Boiling River" near Mammoth, don't jump in the boiling pots that idiots boil themselves in by accident)
And leaving the park via Gardiner, will allow you to head towards Livingston wtth the old Chico Hot Springs on the way to Livingston. EXCELLENT FOOD, Brunch on Sundays is amazing.
The Murray Hotel in Livingston, which Anthony Bourdain calls one of the 10 best in the world, is nice. (I thought it was cool, but I live in Bozeman)
Enjoy.. Look out for Solar eclipse rates anywhere near the eclipse. Go North..