If the iPhone is updated to iOS16.1.1 or later you may have problems pairing this watch.
If the phone is on iOS 15.7 leave it there.
The Series 3 has a very small memory and whilst Apple is reported to be working on a fix you may be advised to completely restore the Watch to clear all the memory before trying to Pair.
NB you may lose any data on the Watch
- Open the Settings app on your watch, then tap General > Reset.
- Tap Erase All Content and Settings, then tap again to confirm.
If necessary do not restore from the offered backup
I also recommend reviewing the steps in the articles below
How to pair an Apple Watch with your iPhone How to pair your Apple Watch with a new iPhone - Apple Support or
Can’t pair an Apple Watch with your iPhone If you can't pair an Apple Watch with your iPhone - Apple Support or
Apple Watch not paired with your iPhone If your Apple Watch isn't connected or paired with your iPhone - Apple Support