Check on San Juan last-minute permits for a small group. Flow's 2,300 cfs right now which is pretty good for this time of year.
But then, why do you need a permit? Are you really dead-set on boating in 105 degree desert heat? If you wanted to come up to Colorado, you could go spend a few days on the Arkansas, boating fun runs with an overnight or two thrown in, then head up for a few days on the Upper Colorado, Roaring Fork, or Gunnison.
Or pick up a Westwater permit (open dates here)
put in at Loma, spend a day or two getting to Westwater, and keep going to Dewey Bridge or Moab instead of floating all the way to Hite. Or float from Swaysey's above Green River UT, to Mineral Bottom if you don't feel like running Cat or have a motor.
You could probably hook up with a Buzzard that's got a Westwater permit too.
Here's the USGS flow page for Colorado
or you could check the AW page here
These are the easy suggestions that don't require a lot of planning, you've got plenty of options....
Have fun, whatever you do!