A rental agreement says that occupation of premises by tenant is?
What you are describing is a holdover tent which is typically significantly higher than the lease rate. Your holdover status is not unlawful in any criminal sense, so you are not trespassing by being there. You are in violation of the lease and the landlord has the option of evicting you, which is a civil legal process. Once you pay and the landlord accepts the holdover rate you are on a month to month status at that rate and the landlord must give you required notice to evict you, either 30 or 60 days depending on the length of your tenancy and the jurisdiction you live in.