The "JMT" permit is just a single permit for all the various parks, forests and monuments the JMT passes through. You can hike with split permits -- just start calling the parks and seeing what you need. I'd expect that the HI entrance permit is the hardest to get, but if you're not planning on exiting at Whitney Portal, it maybe easier.

Near MTR, you can exit to the east side of the Sierra via the Lake Italy Trail/ Pine Creek trailhead at about mile 95, or you via the Piute Canyon Trail to Piute Pass Trailhead at mile 110. Either of those trails is about 16 miles to the trailhead, and from the trailheads you hitch a ride to US-395, then a ES Transit bus to Mammoth and a YARTS shuttle back into Yosemite.

FWIW, I recommend entering at one of these trailheads and hiking South to either Whitney Portal or Horseshoe Meadows. The southern portion of the trail is harder than the northern part, but definitely prettier.

Planning tips -- get Elizabeth Wenk's book and the NatGeo maps and just start reading. Think about food weight, miles per day, and elevation gains. These to blogs have some great info: http://bearfoottheory.com/category/john-muir-trail/ http://jmtbook.com/blog/