Using WordPress With External SMTP Server
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In /wp-includes/pluggable.php around line 377, change
Then, in /wp-includes/class-phpmailer.php around line 155, set your SMTP host:
var $Host = "my.smtphost.com";
You may also need to set a username and password, and tell WP to attempt authentication. You'll see those in the lines below the hostname variable.
var $SMTPAuth = true;
var $Username = "username";
var $Password = "password";
On the other hand, you could do this via a plugin, perhaps even Callum McDonald's WP Mail SMTP.
I found this soloution here