There are plenty of trails that are open to bikes in Jackson Hole.
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Teton Pass is your playground. There are trails going every which way from the pass’s summit, but the truly burly want Phillips Pass, about three-quarters of the way up the pass from the Jackson side. Start up a very dilapidated forest service road on your north (the road is labeled Phillips Canyon). You’ll see signs for Ski Lake, but you don’t want to go there. Continue heading up for a little while longer until you’re into Phillips Canyon proper. From there, it’s all downhill. A nice 30 minutes of rooted, rocky singletrack downhill. You’ll pop out on Fish Creek Road, about three miles north of Wilson. From there, you can ride into town and hitch back to your car, bike up the pass to your car or, with some pre-planning and a shuttle, greet your car at the finish (there is a small parking lot there).