First - have you acquired SSL certificates for the new and the old name?
The SSL certificates should have both the internal and the external domain name in them.
Thus:
mail.example.com (common name)
autodiscover.example.com
server.example.local
server
You would change the addresses to suit your own needs.
Once you have resolved that, you then need to look at the URLs.
Use get-clientaccessserver | fl to view what they are.
You will also need to adjust the web services, Outlook Anywhere and Exchange ActiveSync URLs as well.
The internal URLs should match the server name and that is why it is included on the SSL certificate.
The external URLs can be whatever you like as long as it resolves and is on the SSL certificate.
After making the changes, run IISRESET.
Simon.