Gosh, I have been boondocking in MT for the last five years. Especially around the area you are looking at. There is a dirt rd that runs from Twin Bridges over to Divide, there is plenty area to park for a night or day.There are numerous places around the Big Hole, you just have to look.There are several FS roads along the Gallatin Canyon that you can stop at. I have not found a place that I could not find a place to spend the night. Ruby Reservoir has a some nice shore line. In Ennis if you want to spend the night there is a parking area behind Town Pumps. Along the Madison the are several places that are BLM and FS that you can use some are free some have a small fee..
The same goes for WY and Idaho. Just get yourself a map and figure something out.
There are FS and BLM offices in West and Dillon, those folks are very helpful.
I have some that I use for days at a time while I'm fishing but I'm not gonna share. Sorry but I don't want them over run. Not sayin you would trash them but I have found you tell one, then they tell someone, well you know how it goes.