On May 1, they will release unpaid Whitney permits throughout the day at random times.
http://www.fs.usda.gov/detail/inyo/passes-permits/recreation/?cid=stelprdb5150055#timeline. So that is your next opportunity.
White Mountain, by the way, is 14 miles round trip, up a jeep trail, starting at 11,000 feet. Sometimes, they open the gate, and you can drive to the lab, at which point the hike is only 10 miles round trip up the jeep trail. It's a good hike, although lacking the variety in terrain of Langley.
