Occasionally, an update leaves your messaging settings in a partial state.
Go to:
Settings >> Messages >>
Turn Off Send as SMS
Turn Off MMS Messaging
Wait 10 seconds
Turn both back On
Then test with someone who is not using an iPhone.
Reset Network Settings
(This resolves most carrier-related SMS duplication issues.)
Settings >> General >> Transfer or Reset iPhone >> Reset >> Reset Network Settings
This will not delete your data but will remove saved Wi-Fi passwords, Bluetooth pairings and VPN profiles.
Update Carrier Settings
Sometimes, after an iOS update, the carrier settings do not fully update.
Go to:
Settings >> General >> About
Wait 5 – 10 seconds.
If a pop-up says Carrier Settings Update, tap Update
Turn off iMessage temporarily
This forces your phone to re-register its SMS routing with your carrier.
Settings >> Messages >> iMessage >> Off
Wait 30 seconds.
Turn it On again.