If you are flying into Reno, it’d be most efficient and easiest to take the bus to Lee Vining. Hitch up the hill to the Wilderness Center in Tuolomne Meadow, a place where you can get a permit. If you arrive late, you can camp in the backpacker’s site at TM.
Many “thru-hikers” on the JMT actually start at TM and head south, avoiding the craziness and hassles of the Valley.
When you get a permit, you are really only getting it for a particular gateway and first night camping. After that, you are free to go where you want. So at TM, get whatever portal to the wilderness is open for the next day. Hike in, camp, then hike out the next day and either get on the JMT going south, or for purists, catch the bus for the Valley. Camp in the backpackers campground that night, then head out from Happy Isle the next morning.
So be flexible about where you camp the first night, and it is very likely you can get a permit at TM. You may need a day or two longer to complete your thru hike, but it’s not a race; you are there to enjoy it.
Bryan