TLDR: folks, please check hospital systems that are not NYS official sites. I have at least anecdotal evidence that that is a route to vaccine distribution:
Hey everyone - in addition to state sites (driving far) or job preference (union/hc worker), if folks are in the eligible NY list, you should check what hospital systems are doing. This is a third source of vaccine, which while subject to eligibility rules, gets allocated vaccine and is not a NYS executed sight.
My parents have passed, but after we convinced my inlaws to get the vaccine, they were able to get the first shot quickly without us having any 'ins' by going through Columbia Presbyterian, which releases slots daily.
The 'in' my inlaws had was the resource of my wife putting in time to figure out the best option. Which I freely admit is a valuable resource. My now deceased first generation grandparents would not have had that in the form of my parents, who didn't know how to research, etc., and certainly didn't have the free time to do so.
I was pushing my wife to just get them an appointment in the North Country (maybe 6 hours away), but she found this resource, which was quicker.
Good luck to everyone.