Wind River Range is a terric wilderness full of lakes, mountains, rivers.
Hardly anyone ever goes there, either!!! ;) Depending on the trail and time of the year, might see only a few hundred people in a week. That might be an exaggeration, since I've only gone there after Labor Day -- once the trails have cleared.
Haven't ever used a shuttle to get to, from, or around the Winds, but researched it a few years ago and found a shuttle service that can pick you up at the airport in Jackson Hole, drop you off at a trail head in the Winds, pick you up whenever and where ever you want upon coming out of the wilderness, and drive you back to the airport in Jackson Hole.
Denver or Salt Lake City are probably the closest major airports providing access to the Jackson Hole airport.
Here's a link to the outfit that provides that shuttle service, an outdoor store in Pinedale, WY, about an hour and a half from Jackson Hole.
Drive time from Jackson to Pinedale is about an hour and a half (about two hours once distance from Jackson airport to town of Jackson is added).
Drive time from Pinedale to a trailhead depends on which trailhead. Closest is likely Elkhart (half an hour or so at most); furthest might be Big Sandy (an hour and a half at least). These trailhead drive times from Pinedale are in addition to the drive time to/from Pinedale and Jackson airport.
For us, cost of the shuttle added up to an amount that made renting a car at Jackson Hole airport more cost effective. But if money were no object . . .