Not sure what kind of a vehicle you are in but there is a SWEET road out of Mt. Rushmore. Hwy 16A . If you're in a large work truck you may not be able to get on it.
When I went to Devils Tower I went the back way off I-90. At Whitewood, NW of Sturgis on I-90 I took Hwy 34. At the Wyoming border it becomes Hwy 24.
BTW, I used one of those local tourist maps available at all hotels, tourist centers, etc.
Here are my pics:
Tuesday
South Dakota 2012 Tuesday Photos by mcconned_bucket | Photobucket
Wednesday
South Dakota 2012 Wednesday Photos by mcconned_bucket | Photobucket
Thursday
South Dakota 2012 Thursday Photos by mcconned_bucket | Photobucket
Friday
South Dakota 2012 Friday Photos by mcconned_bucket | Photobucket
On the Thursday pictures you can see a tunnel on hwy 16A.