For permits after late April, the reservations work like this:
"Wilderness Permit Reservations
Sixty percent of wilderness permit reservations become available by lottery 24 weeks in advance.
Of these, any remaining reservations after each week's lottery become available on a first-come, first-served basis the Friday after the lottery application period closes and are available up to three days in advance.
The remaining 40% of reservations become available seven days in advance (up to three days in advance)."[/b]
SO, if you want a late May permit, start looking on recreation.gov at 7:00 a.m. 7 days in advance.
My advice is be flexible on the start date: i.e. if your goal is to start on a Wed, try for Monday or Tuesday, and Thuursday or Friday as well.