Apple Watch connection issue with iPhone

Did you erase your Apple Watch Series 3 completely before starting the paring process?

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1. On your Apple Watch, open the Settings app.
2. Tap General, then tap Reset.
3. Tap Erase All Content and Settings.
4. Type your passcode, if prompted.

5 . If your watch has a cellular plan, choose whether to keep or remove it.

• • If you'll pair your watch and iPhone again, keep your plan.
  • If you're selling or giving away your watch, remove your plan. Contact your carrier if you want to cancel your cellular subscription.

6 . Tap Erase All to confirm.

After it erases, you can set up your Apple Watch again. You'll need the Apple Account and password associated with the watch to turn off Activation Lock.

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copied from: Unpair and erase your Apple Watch - Apple Support

Also make sure that your iPhone and watch can connect to a 2.4GHz Wi-Fi network.

A Series 3 is quite old and may not pair with a newer iOS version at all, so check that your iPhone's iOS is one the watch still supports. If not, that would explain it no matter how many times you reset.