How do I change the phone number of my emergency contact in SOS without changing the contacts? When I try the « Édit » button, it won’t let me edit the phone #, just the person.

Hey Bijoux 906!

You may simple have to delete the emergency contact as you mentioned. This section in the article you came from may be useful:

Use Emergency SOS on your iPhone

“Add emergency contacts

1. Open the Health app and tap your profile picture .
2. Tap Medical ID. 
3. Tap Edit, then scroll to Emergency Contacts.
4. Tap the add button  to add an emergency contact.
5. Tap a contact, then add their relationship. 
6. Tap Done to save your changes. 

You can’t set emergency services as an SOS contact. 

Remove emergency contacts

1. Open the Health app and tap your profile picture .
2. Tap Medical ID. 
3. Tap Edit, then scroll to Emergency Contacts. 
4. Tap the delete button  next to a contact, then tap Delete.
5. Tap Done to save your changes.”

Regards.

Hello and welcome to Apple Support Communities, Bijoux 906!

We understand you're trying to change the number listed for your emergency contact, but it is not allowing you to modify it. We'd be happy to provide information to help you accomplish this.

Try restarting your iPhone and then following the article steps to edit the phone number: Restart your iPhone

Have a great day!