On the Registrar's Student Information Changes form, on the top portion of the form, indicate in parentheses (new) next to the alternate email field. Submit the form with a government-issued photo ID to the Registrar's Office, and they will update your alternate email in the system.
Related FYI: The Registrar's Office and many other offices at Texas Woman's will only accept and reply to email from an official @twu.edu email address.