I use The Eastern Sierra Transit Authority to get from Lancaster, California Metro link station to the town of Bishop to go to my brothers house. From down town Bishop your about 6 to 10 miles away from the trail head. It heavily traveled road with backpacker,fisherman, rock climbers so you could hitch hike or talk to some of the sporting goods stores they may know people that give you a ride to the trail head.
I take the Metro link train from Oceanside,Ca, at 7:00 a.m. To Los Angeles Union station transfer to the Lancaster train to Lancaster metro link station catch the bus at 1:00 p.m. and arrive in Bishop around 6:00 to 6:30 p.m. all in one day.
A little history on Eastern Sierra Transit Authority it took over the greyhound stops in the eastern sierras they run Monday,Wednesday,Friday between Lancaster and Reno ,Nevada and they have stops in every town on route 395 They even stop in Mammoth Mountain. The bus is a small commuter bus and has area in the back that holds cargo like backpacks ,snowboards,skis, bicycle rack on the front of the bus. Fare are very reasonable here's a link to the site.