An alternative to the Snake near Jackson is the daily section on the Salmon river above the confluence of the Main/Middle. The 5+ mile run that starts at the PineCreek bridge is a fun class III run. In July the water is a lot warmer than the Snake (much lower elevation). Downside is an extra 1.5-2 hour drive each way. Upside is a roadside run, no crowds other than a few locals on the weekends, no group reservations required for the run, no traffic control rangers at the put in or take out, big cheap campground (Spring Creek), a variety of run options (including launching from Spring Creek), the invasive species tags are cheaper, and more.
There is also a six mile class III run on the Fall River near Ashton, ID - 4 or so hours from SLC. It can feel a bit remote and is not roadside and has a short season.
What about the Weber river from Henner to Taggart? Less than an hour from SLC, runs most the summer at 400ish CFS. Small river but can be fun in small rafts and IKs.