This is a slightly different approach and starts higher than the south face route described by idahosummits
on the main page
. It involves a bit of bushwhacking but no more than the south route.
The best advice I can give is to use this map:
View Smoky Dome in a larger map
The route starts up an ATV track from the suggested parking area. Head north and after no more than 200 yards watch for a trail breaking across a contour toward Rough Creek on the right. It will intersect Rough Creek where a spring drops in with a lot of foliage and timber above.
Cut up to the right (east) and follow the spring/vegetation up and over a knoll. Lots of sheep sign (we actually saw bear sign) where the vegetation has been knocked down. This is a good path to follow.
Continue to angle up, staying to the right of the trees as you move up and around to break into the open. Continue to angle up the slope to the ridge. The rock tower on the ridge is a good target if you stay to the right of it.
This angle will intersect the summit ridge and the views to the north are outstanding once you crest this ridge. The summit block to the northeast is obvious once on the ridge and the rest is easy.
This route is less than two miles and around 2400' vertical. The south route is closer to 3000' than what shows up on the main page (3700').