Is it possible to extend a visa exemption in Thailand for an additional 60 days, and what is the application process?

Sounds like they are in Bangkok, go 3 days before your current stamp expires. More than likely you'll get 30 days for 1900 baht, then go again just before September 27th to apply for Covid extension.

Yes. Feasible. Some Immigration Offices might ask you to take the regular 30-day extension, first. Others may "directly" grant you the 60-day COVID extension. As for the regular extension, you could apply for it any time you like and it would start counting / be added to the end date of your initial permission period stamped in your passport. As for the COVID extension, the first one would (by most Immigration Offices) be counting from the day you get it - thus you may lose permitted days if you go and apply too early. And with all of that, that is just the rule, but there can always be arbitrary exceptions to it - as nothing with immigration is hardly ever applied consistently across immigration offices (or even officers). Also, be prepared for an "under consideration" situation, meaning, that with COVID extensions, you might initially get an under consideration stamp and have to come back to the very same immigration office about two weeks later to get the final extension stamped in - just another needless inconvenience, but nothing you can do about it. Enjoy your stay - you currently can for as long as you want.

Which immigration office will you use?