Richard, June 3 is a Tuesday. Even though all the permits are reserved, it is so early in the season, half those people won't show up. All you need to do is check in at the Visitor Center (just S of Lone Pine) to pick up your permits the day before your hike (even the morning OF the hike would work this early in the summer). Ask for another permit -- there will be plenty.
Here's how I know: Look at last year's
Unused Whitney Permits numbers.
The Wilderness Permit Options (link in the box above left), especially section 6, gives you more info on walk-in permits.
Get your group ready to go! They will have permits.
(...and if your experience is like so many others, when it comes down to the day to go, you will likely have several cancellations, and won't even need to pick up extras.)