@sjoerdsmeets – did you notice the following note in my article:
Most users on quality WordPress hosting will not need to use SMTP. You should avoid using SMTP if possible as using SMTP can contribute to email authentication issues.
It looks like your new hosting (like many quality WP hosts) allows sending mail from WordPress with the standard recommended wp_mail method – which is good. In fact we would never use hosting that didn’t allow us to do this.
So now there is no need to use SMTP. Which is also good, because setting up SMTP introduces extra complexity, which can be confusing.
If you did need SMTP, you would need to set it up to suit the new SMTP server you are connecting to – your old settings almost certainly wouldn’t work.