If you have a friend wherever you are now, you can take their lease and a faux contract or letter stating you live there as their roommate. I had a roommate do that once without the letter and it worked. I’ve heard others say they needed something from the leaseholder. This was years ago, and laws change so cover your bases.
Someone with better Chinese can check but I think a sample termination letter is here, forth PDF down:
離職證明
https://www.bli.gov.tw/0101272.html
Here’s the foreigner help line. They can answer any legal question, but I wouldn’t mention faux contracts.
0800-024-111
And I think the extension is 6 months now.
Edit: History is repeating itself.
Look at the next posts after mine for a few other ideas.