I did the Tucson to Montana two years ago. There aren't that many ways to get there without going way out of your way. Just use the route Google Maps suggests and hit the road while keeping an eye on the weather. I wouldn't head up there until summer.
If you're going to the Road to the Sun keep in mind the road doesn't fully open until late June or early July. We were in Idaho Falls for the 4th of July celebration (recommended) and then we went to Kalispel after that.
2013 Chevy 3500HD CC dually
2014 Voltage 3600 toy hauler
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