I've mentioned the Grandview campground as a possible place to camp as part of a trip to MW. It's in the White Mountains across the Owens Valley from the southern Sierra. Aside from the Sierra view, the chance for a 4-mile hike around the Bristlecone Pine forest, and even the possibility to bag an easy 14er, there's another reason to build this side trip into your plan.
The Grandview campground has an amazing night sky! On dark (no moon) nights amateur astronomers come from all around for one of the darkest skies and brightest star displays to be seen in the lower 48. If you've never seen a star display away from light pollution it's really worth your time. A Sierra dark night is good - this is incomparable.