I have done a few bikepacking trips in the Wyoming Range and recommend it if you are looking for a true wilderness experience.  BUT, there are very few trail users and the trails are sometimes very faint / non-existant, deadfall and swamps are common, and the insects can be fearsome.  The trails do not get much maintainence by the Forest Service so best to stick to the main trails (Wyoming Range National Recreation Trail).

The trails in the Big Hole Mountains on the Idaho side are more user friendly and also have more peeps out riding dirt bikes.  Awesome views of the Teton Range.

If you want more of a 2 track cruising type of trip, there is a web of old logging roads around Mt Leidy in the Gros Ventre Range that are worth exploring with lots of wildlife to be seen.  Best access from the Blackrock Ranger Station on Togwottee Pass.  Bring bear spray!

There is a huge amount of public land here but much of it is wilderness or national park.
Let me know if you want more detailed info, I've lived here for 20 years--

Carlo

I rode the CT last summer on a steel hardtail 29er but I did add a Thud Buster seatpost to help my fragile lower back.  I think this was a good compromise to a FS bike-- I was really glad for the suspension seatpost in the gnarly sections.  I will second a previous reply in that physical and mental toughness are far more important attributes to a good ride than suspension travel.  I found I had to go really slow on the downhills just to keep the bike from gaining un-controllable momentum and I doubt that I would have been going faster on my FS bike.

It is a great adventure--

Carlo