Ancestry has completed indexing the 1940 census for New York State and is offering free access for a limited time. I couldn’t determine how long that free access lasts. You may want to go to this special web page as soon as possible. You also are going to have to sign up for a free account if you don’t already one on Ancestry. I found most of the people I was looking for including my parents who were not in the town I thought that they had lived in at that time. I couldn’t find my paternal grandfather. It is possible that he got skipped. Another possibility is that the indexers couldn’t figure out the proper spelling of his name. According to the Wall Street Journal, Ancestry used a corporation in China to index the 1940 census.
Also on that special web page on Ancestry is a block on the right of that allows you to search other New York State databases. The best of those databases is the 1892, 1915 and 1925 New York State census indexes that Ancestry recently added. The 1892 census is also on Family Search but this is the first time that the 1915 and 1925 NY census indexes have been on-line. For some strange reason in order to access those records you have to enter a NY State zip code. That is okay if you live in New York State but others are left out unless they know any zip code in New York State. I tried a few different zip codes and they all worked including the zip for Kodak in Rochester; which is 14580 (hint, hint).
Again, I must remind you that these records are only free for a limited time so you may want to do your search as soon as possible.
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Just 15 minutes after the original post, I found my paternal grandfather. He was indexed as Halley instead of as Halsey. I found him by leaving the surname field blank and just including his first name and the name of his county of residence.