Already given the required 60 day notice to terminate

[Mike Hummer]

I served my tenant a 60 day notice to terminate our rental agreement October 3rd, Rental agreement will expire January 4. after that day the tenant will be there unlawfully .

My question is, if she is not out and want to have more time to find another place and offers me rent,  could I make a  one month rental agreement only for the month of January or would that  void out the 60 day notice  that I already served  and have to start over again with a new 60 day notice?

or should I stay the course and file a unlawful detainer after the 4th