I have run the wilderness section a total of ten times over they years, with groups ranging from just one other person up to eight others, and we never had a permit ahead of time. We just camp out by Rand and show up at 7:30am (or was it 7:00am?) when they open. We've always been able to easily obtain permits, and only one time had to come back later in the day to pick up some for whatever number we could not get in the morning.
The caveat: it was 2001 the last time we did it so things may have changed since then, but at least you are a bit closer than we are. If you get skunked, you can always hang out at Horseshoe Bend campground on the No. Umpqua east of Roseburg. Not wilderness, but still very beautiful.
Our "trick" for success has been a) to only show up at Rand Sunday through Thursday, and b) we have never tried it on a Thursday before a holiday weekend. My guess is that your odds would drop dramatically if you ask for permits Fridays, Saturdays, or before three-day weekends.