Here is a snippet that mentions the Social Security Act of 1935 and mentioned amendments:
"The Social Security Act of 1935 excluded state and local government employees from Social Security coverage because of constitutional ambiguity over the federal government's authority to impose payroll taxes on public-sector employers and because these employees were already covered by DB pensions (Nuschler 2021). Beginning in the 1950s, a series of amendments were enacted that allowed state and local governments to enroll certain categories of employees in Social Security. By 1991, over 75 percent of them were covered by the program. Today, public-sector employees are permitted to remain outside of Social Security if their employer-provided retirement plans meet Internal Revenue Service Employment Tax Regulations, which require plan benefits to be sufficiently generous. To meet the generosity standard, a plan must provide members with an annual benefit for life that is at least equal in value to the annual PIA that members would have received had they participated in Social Security. The pensioner must be able to start benefits on or before reaching Social Security's FRA, which varies from 65 to 67 depending on the worker's year of birth."
From:
https://www.ssa.gov/policy/docs/ssb/v82n3/v82n3p1.html