SMTP-Mailer is configured automatically and working for me when I send the test email. However, a number of Contact Form plugins that were advertised as working with SMTP-Mailer weren't actually sending an email. Some failed silently, and some helpfully showed an error to the user.
I did finally find one. Completely free. I'm not connected to them in any way. Just leaving this here in case anyone else runs into the same problem.