Unless they have changed the process since I was out there, they hold back some permits for 'walk-ins'. I was able to get one in the middle of the summer by walking into a station the afternoon before I wanted to hike. I don't know how common that is but it worked for me. (I also got it up somewhere, Bridgeport maybe, where you "can't get Whitney permits", so YMMV)
It's even easier since mconnell was last here, as they now require permit holders to pick them up even earlier the day before (you used to be able to get them from a drop box the night before). There are plenty of unused permits from my experience, even on weekends available starting 11a the day before, I believe (check the USFS website). People love to apply for permits but then other parts of life get in the way and the permits go unused. You have to get them at the interagency office just south of Lone Pine.
I've also heard that it's fairly easy to get unclaimed Whitney permits at the interagency office south of Lone Pine. If you want certainty, I believe the Forest Service on/about May 1 will release permits that were not won in the lottery. I struck out in the lottery five years ago but was able to call on May 1 and reserve a day permit for early September -- it was a terrific hike.