Here in the USA "Persons are classified as unemployed if they do not have a job, have actively looked for work in the prior 4 weeks, and are currently available for work."
source:
https://www.bls.gov/cps/lfcharacteristics.htm#unemp
It's not that you're no longer counted as "unemployed" because you're on welfare - it's that you're no longer eligible (by individual state laws) to receive unemployment insurance. Your benefits have "run out". Each state makes its own laws on how long you can receive unemployment insurance payments.
So when the payments run out - many folks end up on public assistance ("welfare"). And they're no longer counted as "unemployed" ... because they're no longer receiving unemployment insurance payments.
Another big difference between today and the 1930s: today many folks receive unemployment insurance, then public assistance, and sometimes medical "insurance" if you lose your job.
Back in the 1930s the only thing you got after you lost your job: soup.
Tom