This is surely a tough one if the registrant has used a private registration. I would at least call GoDaddy and see if the domain is still within the time frame where you can regain registration control. It MAY be possible that way, but if the time has been too long you may end up having to pay the auction rate. The key variable is how long the domain name has been expired.
Is the domain name still registered at Godaddy? If the domain name is under privacy you are able to send an email to the domain owner by emailing the private email address such as firstname.lastname@example.org
Is it up to the new owner whether to reply to you or not.