How were PUMA and mini-PUMA identifiers assigned to each record in the 5% PUMS data?

I understand how the 1960 PUMA and mini-PUMA geographic boundaries were created, but it is unclear what the methodology for populating these units was. Short of having the specific location of all respondents geo-coded, how did you know within which PUMA each record fell?

Is there a piece of the documentation I am missing?

There is a description of the methodology on the 1960 PUMA and Mini-PUMA page.

To answer the specific question, “how did you know within which PUMA each record fell?”: As part of the 1960 Data Restoration Project, we obtained permission to access, within the Minnesota Research Data Center, the complete 1960 microdata records, which include the county and census tract of residence for all respondents. We then assigned each respondent to the PUMA and mini-PUMA corresponding to their county or tract.