Staying past the ARC date

My school semester finishes on 29 June. My ARC also ends on the same date.

I want to stay in Taiwan for another month after that date.

What is this going to take? Can I apply to stay longer after the date? Or do I have to exit the country and come back in on a visitor visa?

A visa run. However, I think you can simply get a one month landing visa at the airport. The length may depend on your nationality.