Steve D,
If you
check online, there are currently 12 day use permits available in the next two weeks.
Particularly during the two-week period before a hike, people will log in and release some or all of their reservations. So if you keep checking that link, you are likely to find exactly the day you want.
Also, there are always walk-in permits available (free) if you go to the Visitor Center in Lone Pine the day before you want to hike. Check the links in this post:
Unused Whitney Permits for 2014