Okay, I decided not to wait until tomorrow.

No matter how much I loosened the quick release I wasn't able to push the wheel back into the dropouts. The furthest I could get it was about halfway in. And when I did this, the tire was pressing firmly against the left chain stay. I can't tell how far back the wheel sits in the right dropout because the derailleur and cassette are in the way.

Anyway, I have the tire straight now, but the wheel is still in the front 1/4th of the dropouts. I tightened the quick release as tight as possible, but it still scares me a bit with the wheel so far up front. Should I worry about it?