Right now getting to the trailhead for White Mtn is an issue. I drove up to the Visitor Center at the Schulman Grove recently, and had no issues. However, there's a sign posted there that says there's still deep snow on the road beyond that to the trailhead, which is several miles away. I'd call the US Forest Service Ranger Station in Bishop as you get closer - they'll have the most up-to-date info.

Also - do you really want to hike two 14'ers back-to-back? Unless you're incredibly fit, that level of activity is beyond most of us, even when acclimatization is not an issue. As they say, hike your own hike, but ...