In 1960, you will see that the Social Explorer database of the census provides data by County and by "Tract."
Will the print edition provide smaller areas?
Check the Census Catalog and Guide, to determine what statistics were gathered and what was published in the 1960 census.
The Census Bureau Census of Population and Housing page includes pdfs for many of the census years along with information about what is included. You might also try the National Historical Geographic Information System
The National Historical Geographic Information System (NHGIS) provides free online access to summary statistics and GIS boundary files for U.S. censuses and other nationwide surveys from 1790 through the present. Includes some geographies not represented in Social Explorer, above.
Note: In the Census Catalog and Guide, there's a section describing data collected by tract. This includes housing statistics. Tracts were created in order to measure and compare the population below the county level. Tracts attempt to divide the population roughly according to number of inhabitants. Therefore, densely populated areas will have more tracts, sparsely populated areas will have fewer.
Unfortunately, in 1960, only the more densely populated areas of the country are tracted.
To see if your town is included in a tract. go to Social Explorer --> Maps and drill down to your county until the tract numbers and identifying geographical features begin to appear. If you are researching a very rural area of NY, it probably won't be tracted.