There is an ESTA (Eastern Sierra Transit Authority) bus out of Lone Pine to Reno every morning, and a YARTS bus from Mammoth to Yosemite every day. They cross paths in Lee Vining, only YARTS leaves Lee Vining 20 minutes before ESMT arrives. (Go figure!) ...I do see YARTS comments saying they do "stop on demand" requests, so you might make a phone-in request.
You might catch the ESTA bus to Lee Vining and try hitching to Tuolumne.
Or you might leave your car in Mammoth and take YARTS to TM. Then on the other end of your trip, hitch down to Lone Pine, spend the night at Whitney Hostel or anywhere else, and catch the ESTA bus to Mammoth and to your car.
Looking at this thread: Shuttle Partners for JMT on July 31
it looks like the ESTA bus only runs week days.
Good luck, both on the transportation and the hike!ESTA Lone Pine / Reno schedule
ESTA phone#: 760-872-1901 Toll Free: 800-922-1930YARTS
Toll free: 877-989-2787YARTS Hwy 120 / Mammoth schedule