By law, census records must be 72 years old before they are released to the public. The 1960 US Census will not become available until 1 April 2032. You will have to find an alternative type of record.
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I think your best source of information would be the Cook County Clerk's office.
Hopefully they can give you some advice on how to proceed in your search
I found that according to the information in your father's death record https://www.familysearch.org/ark:/61903/1:1:Q2MJ-Q13L
that he lived in an apartment complex which means that he would not have paid
any property taxes so his residency could not be confirmed in that way
unless he owned property in 1960 which could prove his residency.
Federal, State or City Tax returns for 1960 would prove his residency
Good Luck in your search
Dear Ms. Dawson,
Thank you for posting your request on History Hub!
The information you seek must be requested via the Census Bureau's Age Search service. This service provides information about individuals from censuses that are still protected by the 72-year rule, but only to the named person, his or her heirs, or legal representatives. There is a Congressionally-mandated fee for this service. Individuals interested in requesting a search of their personal census records must complete and submit to the Census Bureau a form BC-600, Application for Search of Census Records, or a form BC-600sp, Solicitud Para Busqueda De Registros Censales.
The 1960 records will be open to the general public on April 1, 2032.
We hope this is helpful. Best of luck with your family research!