At Large Camping

We had a large group coming to the Grand Canyon and a bunch of us entered the lottery hoping to get 1 large campsite and 1 small campsite (split between Havasupai and Bright Angel), but only managed to secure 1 large campsite.

My question is about at large camping. Would some of our group be able to camp "at large" for instance somewhere not too far from Havasupai? Or are there only designated areas where you can camp at-large. We were hoping to do South Kaibab to Bright Angel. Are some areas along that main trail not allowed to be camped in at-large?