Here are a few suggestions that might be helpful….
Start your hike in Tuolumne Meadows (TM) and head south. You will miss all the Half Dome and Little Yosemite Campground people and getting a backcountry permit from TM is much easier than Happy Isles trailhead (Yosemite Valley).
If possible, start your hike late in August or even better if at all possible, after Labor Day. You will miss the crowds, get the best campsites, miss the bugs, and get very enjoyable hiking temps during the day.
If you can terminate your hike at Red Meadows (about 43 some miles south from TM) there is a daily shuttle from there that will take you to the town of Mammoth. From there you can stay at the Mammoth Mountain Inn, get a hot shower and some great food or find a local hotel for less money. The following morning you can take a 4 hour shuttle back to Yosemite. I believe it also stops at TM as well. If you are looking for more than 43 miles you can do a loop hike farther south from Red Meadows and back or you can do some other additional loops on the way to Reds and still end there.
You could also start at TM, re-supply at Reds and hike out at Bishop or over Kersarge Pass to the town of Independence, yet I don’t know the exact mileages off the top of my head.
If you start in late Aug or early September, I feel you could start from Happy Isles (Yosemite Valley) and I think you will not be bombarded with the flow of people in the park or at the trailhead. Crowds always appear to die down after Labor day. A hike from Happy Isles to Reds is about 65 -70 miles and you can climb Half Dome if you haven’t yet with a much smaller crowd.
If you want to to hike Half Dome literally by yourself, dry camp at a unmarked campsite about ¼ or ½ mile past the JMT and Half Dome junction. Get up around 4:30 am and hike up to Half Dome. You will get to see a wonderful sunrise and all the lights coming on in the valley. After enjoying the top all to yourself, walk down and continue your JMT hike. I have done this 3-4 times and the earliest I have seen another person was coming down the cables and that happened only once.
I believe there are other locations to come in from the East that are south of TM and I am sure you could design a loop hike of some sorts, yet I do not know the exact mileages on these so I do not want to share too many details for the sake of misleading you with incorrect info.
Lastly, make sure to check the availability of the suggested shuttles as I was last there 5 years ago and don’t know if things have changed or exactly when things shut down for the season at Reds. The Mammoth Mountain Inn to Yosemite shuttle ran daily yet might do weekends only after Sept 1, so you might have to plan accordingly in you want to use that one.
Hope this helps and enjoy your hike.