I have hung below the rim and will again, but I would not recommend it.
In the corridor campground you are not allowed to tie anything to vegetation. So not in Indian Garden, Bright Angel or Cottonwood Campground.
In the threashold areas you have certain designated sites that are not hammock friendly. You might be able to hang here and there, but the more in your party the less likely that you can hang.
There should be no problem hanging on the rim.
It ain't what you don't know that gets you into trouble. It's what you know for sure that just ain't so.
- Mark Twain