That's a tough one, Jeff. But if you are a little flexible, it might still work. What time frame were you planning?
If you study the numbers in
Unused Whitney Permits - 2009, You can see that there are many mid-week times when overnight permits are available. If you show up mid-week, especially mid-june to mid-july, everyone should be accommodated.
And remember, those unused numbers are at the end of the day. Show up at the 11 AM time when the no-shows become available, and your chances are better. And I believe if you are there earlier, as groups come in to pick up their reserved slots, they often shed a few unused spots.
Early bird gets the worm. 
