Was fired this week. Had an interview this week and didn't tell them. How to proceed?

I had an interview on Friday. The resume I submitted for the position indicated that I was still employed at my old company because I sent the application while I still had the job. When I had the interview, I spoke in past-tense about my job, but no one asked about my employment status. What can I do to prepare myself for the truth to come out in the background check?

I think I'll send an email in the morning saying that the resume I used was outdated, and that's all. The new resume that I will attach will show that my employment ended in July 2019. Is this good enough? I want to do this without making myself look bad and also without blatantly announcing that I'm unemployed.

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