Maybe put in fake whois information for the day that the transfer is happening. Once it's moved, re-enable the privacy. The info doesn't even need to be fully fake, you can use a throwaway email address so that you're still reachable.
Any change you make to your contact details will lock your domain for another 60 days. That means, you will be stuck for another 60 days before you can transfer the name away from Godaddy.
http://help.godaddy.com/article/418 - Browse to the bottom and you can see it.
