If you launch on Brown's after about 3 pm you'll have the river to yourself. Better yet, launch at Wilderness Aware ($3 fee) and start with the Milk Run. Float down and find a campsite below Ruby Mt. Campground on public land. Continue to Stone Bridge the next am. If you get on the river before about 8:30 or so, you may be able to stay ahead of the train of rubber buses...
Nothing in the world is more yielding and gentle than water. Yet it has no equal for conquering the resistant and tough. The flexible can overcome the unbending; the soft can overcome the hard. - Lao Tse