It sounds like you're on the right track. Pointing the local Autodiscover record to the new email server and making sure the certificate matches the DNS name (like mail.domain.com) is exactly what you should be doing for Outlook and Exchange to work smoothly.
If things still aren’t working quite right, it might be worth double-checking a few other things:
Make sure autodiscover.domain.com resolves correctly, especially from outside your network if that’s relevant.
Check the internal Exchange URLs (like Autodiscover, EWS, OAB) to make sure they match your setup.
Try running the Microsoft Remote Connectivity Analyzer—it’s super helpful for spotting configuration issues.
And of course, make sure users have restarted Outlook so it picks up the changes.
Let me know if you’re running into any specific issues—I’m happy to help!