Might not be the worst case and I may have missed a small logging road but if you were to head roughly south parallel to the John Muir Trail from around Tuolumn

Might not be the worst case and I may have missed a small logging road but if you were to head roughly south parallel to the John Muir Trail from around Tuolumne meadows it would be about 130-140 miles before you reach a road or US-395 around the Kennedy Meadows store. I think you can just miss the roads around Mammoth and Devil's postpile and then the Road's end trail head in SEKI.