Do I need to transfer my visa to a new passport before leaving Thailand?

From Bangkok it's way better to fly to Udon Thani on an early morning flight and take the mini van at the airport to the border in Nong Khai, cross into Lao and get to the consulate. You can do it in plenty of time to get there before they close.

The next afternoon you can get your passport, get back to the border, cross into thailand and go back to the Udon Airport in time to catch an early evening flight.

That's the fastest you can do it.

@Alas****

If you're going to get a 30 day extension at the immigration office on the current 30 day visa exempt entry you have you might as well get the stamp transferred and get the extension in your new passport though.

You have a 30 day visa exempt entry stamp NOT a visa on arrival ;)

Be that as it may, you can just present both passports when you exit the country.

Make sure you don't toss out the letter that the consulate gave you stating you got a new passport. Sometimes they ask for it at the border