After having a recent COVID exposure, I had no trouble getting a same-day appointment at the SUNY Oneonta testing site which is now allowing walk-ins with expanded hours of operation on Mondays, Tuesdays, and Fridays.
New York State planned ahead and has set up state-run testing sites with more planned in the very near future. One such site that has been chosen is SUNY Oneonta. It opens to the public this Friday, January 7.