Cox doesn't have to upgrade their system to work with the new outlook. As long as the new outlook conforms with RFC specifications for Simple Mail Transfer Protocol, (SMTP), then it will work without issue.
Before doing the below, ensure all email sitting in the OUTBOX is deleted.
Ensure your settings are correct, SSL, port 465, and "My Outgoing Server Requires Authentication" has to be selected. OR. Use TLS, Port 587, and again, "My Outgoing Server Requires Authentication".
As long as your settings are correct, then it will work with Cox, since Cox's servers conform to RFC specifications.