TN,
The 40% reserved for walk-ins are available for pick up at 11:00am the day before your trip starts (there may be exceptions that I don't know about). If there are any available on the day of, you'll have to participate in the 8:00am lottery (held if there are four, or more, people waiting - usually during the summer, there are). If there are any remaining after the lottery, they become first-come, first-serve.
Since you're going for a mid-week start, I'd say your odds are pretty good. (I never failed to get a mid-week permit.)