Here is the recreation.gov site to use to watch for permits for that date: Day Use permits October
Bookmark it (better, make it your "home page"), and keep checking.

Otherwise, no-shows come available at 2 PM the day before (on Oct 8) If you are there at that time, you are sure to get a permit. For that matter, any time after 2 PM should be ok, since in October the quota is almost never used up.