Giving your P-60/P-45 to new employer

I'm shortly going to be let go from my fixed term contract job with a multinational as the role is being migrated out-of-country. I am worried about getting a job after being made unemployed, as , as everyone says, it is easier to get a job whilst still being employed than when you are unemployed. If I said in an interview that I was still employed, what are the chances of a hiring manager in a multinational finding out that I am actually unemployed? If ( big IF) I get another job, it will almost certainly be at another multinational who would either have a payroll department or an outsourced payroll bureau running thier payroll side of things. Presumably, I would give my P-60 ( or P-45) to a payroll deprtment, what are the chances of a hiring manager seeing it, or do they normally want to see sight of it anyway and ask that you give it to them?