CSF features great option SMTP_BLOCK which block outgoing SMTP for all users except root, exim and mailman. I had a problem with one user which was using MailChimp as mass mailing within their application. Because of SMTP_BLOCK it wasn’t working. Disabling SMTP_BLOCK globally is not recommended, you can white list users for which you would like to allow sending.
Go to your CSF settings and find SMTP_ALLOWUSER. Then add user which should be allowed (users separated with coma). Don’t forget to restart CSF.