I take it that you have a re-entry permit? I can't see any reason to transfer it as you're not renewing. The only problem is the IO on exit may make a fuss.
The 60-day entry is not a visa it is a visa-exempt entry. As you are returning so quickly, I suggest you book an onward flight within the initial 60-day free entry, a hotel booking for the first night and carry 20,000 baht in cash. This is in case you are selected for secondary screening due to repeated entries. You should get a Destination Thailand Visa, which gives you 180-day entry and lasts 5 years. You can book an onward flight for about $14 - there are many websites but the ticket is only valid for 48 hours so book just before you travel.
I'm assuming you had to make an account to book the appointment. If you did, make another account with a different email and use that account to make the new appointment.