Background check advice for contract job

I had/have a contract role that was renewable by quarter, with the team letting us know if they are going to need us each quarter. It hasn’t been renewed recently and doesn’t seem likely for Q3. My resume still says "present", but concerned about background checks at offer stage. It can still be renewed, but I’m not on payroll right now. Worried about implications when HR runs checks (if not renewed by then). Didn’t expect it not to be renewed.

This has been making me hesitant to even apply to new firms. Could use advice.