Most of the fins on the east side of the Gallatin are orientated E/Wish, so the northish facing sides get shade until about 3-4 pm. The south faces start to get shady around then but they are radiating heat! I would recommend Norther Exposure for the coolest, shadiest wall, with some great climbing as well (Over Exposed, Flight Risk). Its a grunt of a hike, however. High Gravity, Nu Ethix, backside of Wafer (no much info on MP, but in the book), and Tango are all shady and have multiple routes worth doing.
There is a small limestone crag that with five or six 5.12s that stays shady all day at red cliff campground (energy wall on MP). Not only is it shady but tends to be a bit cooler than the fins further down the canyon, due to proximity to the river. its another 20 miles south from the 35 mph bridge but has a very short approach.