There appears to be an immigration office on Koh Tao Here's the link to Surat Thani immigration:

Contact Suratthani Immigration

No idea if you can do extension at that office or not. If nobody here knows, you can try one of the thai visa facebook groups. It's not likely you would waste an entire day there. You go early when it opens and you will probably be out in relatively short order compared to my experience in Bangkok, which is crazy busy.

-David