This is a great thread. A resupply near the midpoint is possible using Sequoia Kings Pack Train on the JMT at the Kearsarge Pass junction. They only require a 1/2 day to get to that point so the billing is only a one day use of packer and mule. I've used their service 4 years now. (I do the JMT annually, will be doing my 6th this summer. I've done the High Sierra Trail for 7 years prior to that.)
Another option is you can do the High Sierra Trail as a lasso-loop round trip that is around 150 miles long, nearly the same length as what you show. It enables a resupply at Lone Pine, via HorseShoe Meadow Trailhead (less costly than use of the Packers at the Kearsarge Pass junction). I've done it. Great hike.
Start in Crescent Meadows in SEKI.
At Kern River meetup on the High Sierra Trail (HST), go downriver, not upriver (as the HST states), go downriver 9 miles to Kern Canyon Ranger Station where there is a suspension bridge over the river.
Cross river, go east about 18 miles to Horseshoe Meadows TH, resupply via Lone Pine (Whitney SHuttle Service or hitchhike from TH to Lone Pine), stay overnight in Dow Villa, enjoy hot spa, and get back to Horseshoe Meadows TH before 24 hours has elapsed to avoid losing your permit. Get back onto trail but head north on the PCT up to Crabtree Meadows. From there go up to Whitney Summit and Back, then head back to Crescent Meadows via the normal HST route, and you'll get back to the Kern River junction where you originally went downriver, but now you'll stay on the HST to get back to Crescent Meadows. Great hike. You'd need two maps Tom Harrison SEKI/Whitney high country map and the Golden Trout Wilderness map from Tom Harrison.
Advantage of this hike is that you end up where you started, easier to deal with logistics. There is a shuttle bus from Crescent Meadows to Lodgepole, and from there to Visalia, CA where there is a small airport or buses to major airports.