It's best for you to go to the nearest Thai Embassy or Consulate in your country and deal directly with them so that you can get the right and proper details for your matters on visa and so..
The moment you stamped out of Thailand, your 60days visa is invalid. When you re-enter, you will be getting the normal 30days visa exempt before 1st October or 45days visa exempt after 1st October. Should you wish to stay for another 45days, then you can apply for it with a fee of 1900bahts as long you can provide a valid reason for the extension