There are some good campsites near the saddle with some great views of half dome.

Better yet hike in from Tuolumne meadows for a couple days. Hike over clouds rest.

Previous poster is correct. you can get a permit for any trail and just throw it away after staying at the backpacker's camp the night before your half dome permit. But I'd look for one a to start 2-3 days before your permit and backpack from Tuolumne meadows. Hike to half dome and down to YV and catch the bus back to your car.

Check the mosquito report in June before picking where to start from the meadows or yosemite valley are ~3500 ft of elevation difference so depending on the temperature and snow melt the mosquitos can be great at one site and terrible at the other.