I like the campgrounds south and west of Jackson on the Snake. They are in the canyon between Jackson and Alpine and are usually not crowded. It's an hour or two to the gate of Yellowstone, but the park is jammed from now until mid September. I live 3 hours from Yellowstone but avoid it most of the summer because of the crowding. Lots of tour buses and pushy people that like to get too close to the wildlife. Last week a couple of guys put a bison calf in their minivan because it looked cold. This week a guy fell into a hot spring and hasn't been seen since. Last year a woman went airborne while taking a selfie with a bison bull. It's always something!
I plan on taking lots of trips in my new camper and will post any good finds promptly. I will say right now that a trip to Stanley, Idaho is something that everyone should consider. The Sawtooths are awesome and every bit as scenic as the Tetons. Lots of great camping and boondocking opportunities there. Several lakes that mirror the mountains and all have campgrounds. Camping right on the Salmon River is where I'll be this weekend slaying the trout.
Yellowstone is a wonder, but can be a traffic jam. Be sure to plan a few days in central Idaho if you can. It is the Rockies without the press of humanity. If you need a quick overnighter near Pocatello PM me and I'll detail a few spots. Lots of great places to stay right off I-15 on the way from Salt Lake to Yellowstone. And take US30 and ID34 going through Soda Springs to the south end of Jackson -- it's just gorgeous!