When can I apply for a 30-day extension on my visa exempt entry in Thailand?

On most Immigrations you can apply when half of the stay permit is left, that would be 30 days before expiry in your case.

You need to be properly TM30 registered in your accomodation by your hotel or landlord, to be able to apply for the 30-days touristic extension, on most Immigrations.

You need your passport, copies of the entry stamp page and the page with the personal information, signed in blue ink, and 1900 Thai Baht application fee.

And most probably since May 1st this year, a printout of the TDAC online form.

The application form TM7 can be had and filled out at the Immigration office

The process is easy. Depending on the office you use will determine how fast it happens. In Bangkok probably a few hours. My guess in UT would probably be 15 minutes or so.

You can usually apply within 30 days of your current exit date (office dependant)

Apply early. They add on the 30 days to your existing stamp so there is no downside to doing it early.

I have no idea of the policy of UT immigration but most offices will allow you to apply 30 days prior to your stamp date (some allow 45 days).