If you are willing to allow dirt roads in your definition of "getting away", you can get a hell of a long way from civilization in the western US.
A few places come to mind:
The Deep Creek Range of western Utah (simply stunning)
The North Rim of the Grand Canyon SSE of Hurricane, Utah. Not the popular tourist areas but the extremely remote canyon rim around SB Point.
The Toiyabe and Monitor Ranges of Central Nevada. Pick any valley in the Range-and-Basin country, drive halfway down it, and then head east or west into the mountains.
The Owyhee Desert. Google for Crutcher's Crossing.
A few places come to mind:
The Deep Creek Range of western Utah (simply stunning)
The North Rim of the Grand Canyon SSE of Hurricane, Utah. Not the popular tourist areas but the extremely remote canyon rim around SB Point.
The Toiyabe and Monitor Ranges of Central Nevada. Pick any valley in the Range-and-Basin country, drive halfway down it, and then head east or west into the mountains.
The Owyhee Desert. Google for Crutcher's Crossing.
The Henry Mountains of south-central Utah.
The Magruder Corridor of Northern Idaho.