Some enterprising individuals have tried to take the word-of-mouth system and digitize it: Dr.B: "The standby list for leftover COVID vaccines."
It is a new approach for getting vaccines that are about to expire into the arms of willing recipients. Likely it doesn't have a totally national footprint because it is so new, but as word spreads there will be more providers who sign up to participate, which eliminates them becoming a "call center" at the end of the day looking to get people to come in for shots when others have not shown up. It is another solution.
Trying to register and get scheduled on state or other websites can be frustrating. My wife and I are glad to be done with "the hunt."