Jon, looking at last year's
unused permits, there were over 50 unused on Saturdays in June. I think your chances of getting the walk-in are pretty good.
Those walk-in permits are free. But if you want to get a reserved one, keep checking
this link, and one might show up. Watch especially in the week or more before your hike. If it does, then jump over to your existing reservation, and modify it by increasing your group size.