32c might do it as long as the tires are not cheap. I rode 37c.
If you want to take a nice detour from the route with some dirt, consider this: East of Jackson you will cross Big Hole Pass. About 6-7 miles down the other side, you will see a brown and white sign for the Pioneer Mountains Scenic Byway. Hang a left there. Very nice road that takes you up to nearly 8,000'. Most of the climb is pretty gentle at first. It's really only the last 5 miles that pose a challenge. Then there is some 26 miles of mostly descent to one degree or another. It's a very pretty ride that includes a beautiful mountain meadow zone. The road ends in Wise River, where there is a bar/restaurant and a small but adequate mercantile for groceries. From there, you head east on MT 43 through a cool canyon carved by the Big Hole River. About 9 miles east of Wise River, just before you cross the bridge over the river, there is a gravel road leading to Divide Bridge Campground. (Go past the day-use area to get to the camping area.) IIRC, sites are only $6 or so/night. Good water and pit toilets. You can wash off in the river. The next day, continue east on MT 43 a few miles until you get to the frontage road for I-15 and hang a right towards Melrose. You can have breakfast at the Hitching Post in town. There is also a motel and campground there. On the south end of Melrose you will see a sign for a left turn to Twin Bridges. That's Melrose-Twin Bridges County Road (a/k/a/ Melrose Bench Road.) Twenty-three or so lonely dirt miles later you will hit pavement again just outside Twin Bridges. After you cross the river, make a right at the T to get to the center of town. If you see three steer (which is likely) you will likely see more steer than motor vehicles. You will also encounter some good ups and downs, and some portions of the road can be rough. Jackson to Divide Bridge may seem like a long way, especially with two passes, but it's not as hard as the mileage suggests. Once you get to the summit, the way into Wise River doesn't require a whole lot of effort. And Wise River to Divide Bridge is a net elevation loss. If you have the sort of tailwind I had the first time I did that stretch, it will go by in a flash. Divide Bridge to Twin Bridges is short (only about 35 miles), but it would give you a chance to rest up at the Bike Camp, which is very relaxing, and check email at the library.
IMO, this route is more interesting than staying on the official route. It also avoids a stretch of road between Dillon and Twin Bridges that has no shoulder and some truck traffic. The downsides are that you miss the town of Dillion, which has many services, and you miss Beaverhead Rock, which is a significant Lewis & Clark related monument.
