The facebook group 'New York City Genealogy' is a good place to ask this sort of question - they know about some of the more obscure places to look for records for NYC residents. They also know what sort of medical records are available for public viewing, or how to go about inquiring about records. The same for arrest/prison/court records.
I would also check to see if there are phone/city directory entries for the grandfather you are looking for, and also look for any other type of record, like voting records (I've found these on Ancestry for a relative who lived in the Bronx in the 1920s) that might place him somewhere in the 1940s.
Lak/Lok/Liak/Lock and Kalon/Kolon in Zagare/Joniskis/Gruzdziai, Lithuania
Lak/Lok/Liak/Lock in Plunge/Telsiai in Lithuania
Trisinsky/Trushinsky/Sturisky and Leybman in Dotnuva, Lithuania
Olitsky in Alytus, Suwalki, Poland/Lithuania
Gutman/Goodman in Czestochowa, Poland
Lavine/Lev/Lew in Trenton, New Jersey and Lida/Vilna gub., Belarus