One thing about reservation systems, a lot of people reserve and then drop out. So there are some permits if you are willing to wait to get a "no show". Best to avoid busy season. We went in November once- it snowed on the rim and many who had permits bailed out. We got a permit for the next day and also got a campground spot, because the campground campers also bailed out! By the time we got to the bottom it was a nice 75-degrees.

Not sure how things are now. You should call the rangers and talk to them about their current "no-show" policy.

It is very hard to get a reserved permit to climb Mt. Whitney here in California- it is a lottery (cost $30 too). I simply wait until after Labor Day and am flexible- and every time I get a free no-show. It may be for the same day, so you have to be ready to go. Or it may be the next.