I lived just south of JH in Star Valley (in Etna on the north end) from 2006-2015. Best suggestion is to download the MVU maps for the national forests in the area (Bridger-Teton, Targhee, etc). There are tons of forest service roads in the area that'll take you pretty much all over the state, and into and thorough Idaho as well, on dirt if you plan it right. I was a Sheriff's Deputy in Lincoln County when we were there (retired now) and did forest patrols in the summer, so I got pretty familiar with treks down the Wyoming Range while getting paid to do it on my days off, lol. One of my favorites was from Alpine down to Labarge and Kemmerer over the course of a weekend with the family. There is also a several hour drive (if you take your time) down Grey's River Rd. from Alpine to Afton that you can use to get your feet wet, so to speak. Plenty of disbursed camping along with some developed campgrounds if you want those too. About the only drawback is that there is practically no dirt to be had in either Grand Teton or Yellowstone, so those two are pretty much out of the question.