How can I extend my 30-day exemption stay in Thailand without applying for a visa?

You can as explained only extend your visa exemption one time and get additional 30 days on your visa exemption. The only visa you can apply for in Thailand is a Non O visa. So if you want to stay for 60 more days in Thailand after your exemption runs out, you don’t have any other options then get another 30 days visa exemption.

If you are sure you entered on visa-exempt for 30 days, not on visa-waiver, you can apply for a 30 days tourism extension ONCE at your local immigration office <- that means the office services your current address on your latest TM30 in Thailand.

This is general information for tourism extension, may vary by the immigration office. Check with your local office for their specific requirements.