YardstickAngler: "Nearly all BWCA sites that haven’t been badly burnt are suitable for hammocks. Especially if you’re the only one hanging. Do it!"
With a little looking you can find maps that shows areas that have burned. Most of your route is good, but there are a few sites on the north end of Little Sag and north side of Gillis that were part of the 2006 Cavity Lake fire. However, in my experience there are sometimes hammocks spots even in campsites that have burnt.
Another consideration is how many are in your group? If it is a larger group that needs a number of tents that can take away some good hammock spots (many of the good hammock spots are across tent pads). Also, are you talking about peak summer season or shoulder season? If the former, you may not have many campsites to choose from but if the later you can keep moving until you find a site that works.