Seems like we had someone at the fly shop in Cody who knew someone to shuttle. We had two rigs when we were in Cody and didn't need to, but I think if you check around you'll find someone. The North Fork was too woody when we were there, but looks fun and we did the town section of the Shoshone and it was very fun. The kids liked that better than the upper Snake. Plenty of opportunities for shuttles around Jackson. The canyon was like outfitter city with a bus pulling in every 10 minutes. Fun whitewater though.
If you lean over to watch the river slipping slowly away beneath you, you will suddenly know everything there is to be known. Winnie the Pooh