Kings Canyon National Park
May through October
I've backpacked in the Kings Canyon area twice, in May and September, which are what I would consider the bookends of the season. Any colder, and this would be for serious Winter lovers only. In May 2016 (Memorial Day), we climbed to about 10,00 feet in the Golden Trout Wilderness, and though there was some light snow on the ground , we were still able to find dry spots for the tents. The small alpine lakes were not yet frozen over. We spent a comfortable two nights there.
On my first trip in September 2015 (Labor Day weekend), there was no snow at all. Perhaps surprisingly given the holiday weekend, we encountered very few fellow travelers, and spent a night completely alone in White Chief Bowl. The weather was perfect for backpacking - just a little brisk.