You can't actually transfer the visa, you will need to keep and show both passports upon entry and exit until the visa expires, the stamps will be put in the new passport.
Did you enter Thailand on a 60 day visa that you bought at a Thai embassy outside of Thailand, or did you get a free 60 visa exempt entry stamp when you entered Thailand? If you bought a real visa and didn't get a re-entry permit before you left Thailand then you don't have a visa any more.