What paperwork is needed to transfer a NON-O stamp from an old passport to a new one?

Every immigration office has its own rules. In Hua Hin I just needed my old and my new passport and I had to fill on a form. That's all. Of course, I also needed to pay a fee. I think about 500 baht. The whole process took about half hour.

That is not true! I got my retirement extension in Pattaya, I went to USA for a few months for a surgery, came back to Thailand 2.5 weeks ago and got my stamp transferred in UdonThani into my new passport. 500 bht. Also got new 90 day check in in UdonThani.

Here's the general requirement and form for transferring stamps from an old passport to a new one.

/Transfer-Stamp-to-New-Passport-Form.pdf

in case Immigration wants to see a certified by your embassy document telling them that you have been issued a new passport, try to evade it by politely saying that your embassy does not issue such a certification anymore (a new passport is a new passport!)

you need nothing but your old passport and the new passport. And maybe the printed TDAC to show (if it applies). Immigration will transfer only their own stamps - which in your case would be the "extension of stay permit" stamp and the re-entry permit stamp (in case you bought one). The "visa" itself cannot get transferred, the visa itself is a separate .pdf document. And if you entered on the new passport, the entry stamp will already be inside the new passport