Greetings moos165,
Thanks for visiting the Apple Support Communities! I understand you were logged out of Game Center after resetting your iPhone. Now, you're curious how to recover your Game Center account, as you're unable to sign in to Game Center.
To sign in to Game Center, tap Settings > Game Center, then sign in with your Apple ID: Use Game Center to play with friends - Apple Support
If you're receiving an error when attempting to sign in, some troubleshooting steps to take are as follows:
If further assistance is needed, it would be helpful to know the behavior you experience when attempting to sign in to Game Center, including any error message that is received.
Cheers.