How can I change to a new passport while in Thailand with a 30-day visa stamp?

Technically the stamp can be transferred at local Immigration, practically most Immigration offices only bother transferring visa extensions and not visa exempt stamps, and they would tell you to just sort it out upon exit from Thailand, when you will get the exit stamp in the new passport with an adnotation of where it's coming from. So just have both passports handy when you exit.