Using the smtp setting to send email through an smtp server did not work on my installation. I had some warnings in the error logs. I found that the SMTPSecure property in the PHPMailer object was not set. After fixing this sending mail through smtp went fine.
I made the following changes in custom-contact-forms-front.php:
Added the following line on lines 419 and 570:
$mail->SMTPSecure = $admin_options['smtp_encryption'];
I hope this fix can be added to the next release.