Definitely check with your eviction attorney but in Texas if a tenant in eviction applies for assistance to get their free 60 day abatement card and their lease has already expired you can file a 30 Day Notice of Holdover (eviction of holdover tenant) which can cut about 30 days off your eviction timeline because you are no longer filing to go after unpaid rents.
You are evicting to get possession of the unit because the lease expired which, per my Rockstar PMC, this overrides the 60 day abatement.
P.S. In some jurisdictions you can also ask for a new judge if the current one is dragging their feet or constantly siding with the tenant.