Here's a recent news release by the New York State Library about its digitized copies of the statistical censuses that where done by New York State in the years in between the federal ones. In many cases if the publication was long we have split it into separate volumes. For example, the 1875 Census was split into 5 volumes. The last one ever done was in 1925.
The New York State Library has a new web page that highlights and links to digitized materials relating to the Census of the State of New York. The State Census volumes that have been digitized are the published statistical schedules compiled from the original returns under the direction of the Secretary of State which describe the population and economy of counties, cities and towns in New York State. Instructions for the enumerators taking the census have also been digitized, as well as, other related documents. These documents are freely available online as PDF documents on the State Library's website.
Note - Information in the Instructions for enumerators can sometimes be helpful when using the State census schedules that list acutal names.
For example, this paragraph below is from page 19 of the Instructions for Taking the Census of New York in the year 1855 ......
8. Color.1 the person be white, no entry is to be made, but if mulatto, write M. ; or if black, B. opposite to the name. In like manner if the person be an Indian, living apart from any tribe, and not on any reservation, write Ind. The number of Indians of this class is probably extremely small. Special marshals will be appointed for taking the census of Indians residing together in settle ments, and the marshals appointed for taking the census of the election districts of towns will have no duties to per form in relation to such Indian settlements. Special marshals will be appointed to take the Allegany, Cattaraugus, Oneida, Onondaga, St. Regis, Tonawanda, and Tuscarora Indians. All others should be taken by the town marshals.
If you're interested instead about the original returns with the names, the New York State Library's holdings on microfilm can be found HERE.
A broader list of some of the items digitized by the New York State Library can be found HERE.