How enable PHPMailer to write complete persistent log?
It's not clear from your question what Drupal version you are on. But the Maillog / Mail Developer module might be handy here. Some more details about it (from its project page):


  ... provides an easy possibility to log all Mails for debugging purposes. It's possible to prevent the mails to being sent, so there is no need for an extra mail server to test the mail functionality of other modules or the drupal core. Additionally you can immediately display the mail through the Devel dpm() facility.


But you might get integration problems, since it's not easy and highly Drupal version dependent to get more than one email module at a time working.