A slightly younger neighbor than I, who is not quite old enough to be in the eligible cohort, said she went to a mass-vaccination site on Super Bowl Sunday to see if they could take her because of possible no-shows. She said she was given her shot within minutes.
But that's not a very useful trick, since waiting until next Super Bowl Sunday may not sound all that great.
But here is what I would recommend to everyone eager to get theirs. If you are near a mass-vaccination site, go down and just talk to the people working there. I did that, talked to the workers who were almost all Army enlisted, and while they said they couldn't squeeze me in, they recommended that I go on the official website and set an appointment. When I pointed out to them that I was a few years shy of the qualifying age, they said it wasn't a problem.
So I went to the scheduling website, put my honest date of birth down, showed up, and voila, I got stuck a couple of days ago. Otherwise, I probably would have been waiting weeks. So just go down there and talk with them. YMMV, but that's the one weird trick.