My favorite floats are from Kountz Bridge to Cardwell and Cardwell though the canyon. From Kountz bridge to Cardwell the river flows though cottonwood bottoms with a lot of braiding, islands and channels. It is away from any roads and although a few homes are visible on the trip, generally signs of human habitation are absent. On this float one can expect to see eagles, osprey, herons, whitetail deer, lots of waterfowl, pelicans, and I've even seen trumpeter swans. There are no dams, fences, or hazards other than an occasional sweeper. Mayflower Bridge provides an access point about 2/3 of the way on this float if you're pressed for time.
The section from Cardwell through the canyon has remarkable limestone cliffs towering above the river. The last time I floated this there was a nesting pair of Golden Eagles near the entrance to the canyon. Unfortunately, a road parallels the river at this point but it is lightly traveled and not intrusive. There are several possible takeout points in this section.
You will have to arrange your own shuttle. The access points are spaced well enough that a mountain bike is a viable shuttle vehicle if you want a few hour's float time and can't get a partner.
This river is 83 miles long, all of it floatable without the crowds seen on the Blackfoot or Missouri.