I've always wondered about how this part is determined - do people somehow "report" that they've given up? If one doesn't have a job, can't find a job they don't necessarily give up completely, do they? They surely would be open to a job, one would think. Besides, what alternative is there?
There is another survey called the business survey and this one determines the number of jobs added or lost during the month.
So it is possible that the household survey can suggest that jobs were added in the past month because it captures people starting businesses from home, etc, while the business survey indicates many less jobs added since this survey does not capture any new businesses started. The surveys quite often conflict with each other, but over a longer period of time, they come together.
The household survey quite often captures a recovery faster than the business survey. The business survey will catch up over time