Mike, permits show up online, especially next day on a pretty regular basis. You just need to keep checking. It sure seems like you could snag one before starting the trip. True, though, there are lots of alternate trails if you go up and can't get the permit.
> In the event I see a permit for a day that works but it is for an overnight, can I use that even if I'm a day hiker?
The wilderness permit people frown on that, but you wouldn't be cited for hiking without an overnight pack.