Mt. Rainier, in a dry year, is very marginal for late June/early July. This is NOT a dry year!

The Grand Canyon and Zion will be at their hottest.

With all of these, check the timeline for getting permits. You may already be too late.

Lower elevation valleys (Hoh River, Enchanted Valley) in the Olympics may be a possibility. However, at least in the Pacific NW, there is lots of trail damage.

Late Winter/Spring hikes in Oregon. Ignore the one about the Wenaha River; it burned badly two years ago and no trail repair has been done. By July, the river-level trail In Hell's Canyon may be as hot as its namesake.

Early/late season backpacking in Colorado

As I mentioned in a similar thread (see General Discussion section), much of the best backpacking in the West is outside national parks, with generally less crowds and less bureaucracy. Of course the same limitations for altitude and snowpack--above normal in the sections of the Rockies with which I'm familiar--apply. Frankly, this might be a good year to defy Horace Greeley and "go east, young man," unless you can come in August or September.