My new iPhone will not pair with my Apple Watch

This could be a compatibility issue, if your iPhone is already running the current iOS 14. Apparently that version is not able to complete the update for a backlevel watchOS 6 any more. If that seems to fit your situation, there two options come to mind:

• Find an old iPhone with iOS13 or older on it and complete the pairing and update to watchOS 7.5 with that. Once that is done, you can re-pair the S3 with your own iPhone again.
• Contact Apple Support or visit an Apple Store and ask if they can run this update for you.