The popular class II/III run on the Snake River is often referred to as Lower Canyon. The run is 8.5 miles long, with put in at West Table Boat Ramp and take out at Sheep Gulch Boat Ramp. Sheep Gulch boat ramp is only about 3 miles up the canyon from the town of Alpine, WY.
On weekends the run tends to get really busy with a lot of commercial raft trips and usually plenty of kayakers. The weather in early June might be iffy - which could lessen the crowding. It's easy to find a group to join w/ on weekends, less so on weekdays. If you don't run into anyone at the put in or take out, you may find a group at Lunch Counter or Kahuna parking/playing. This is easy to spot, a good sized pull off/parking lot on river side of the road about 1/2 way between the take out and put in, w/ a trail down to the river. If there are a few cars there - likely a few kayakers down at the river.
FS website listed below for some additional basic info.
Bridger-Teton National Forest - Teton Division