Which Elgg version are you using?

The latest version of Elgg have the config option for smtp in settings.php in the elgg-config folder:

/**
 * Configure emailer transport
 *
 * This setting can be used to select a different emailer transport. By default the Zend Sendmail Transport is used.
 * Currently only 'smtp' and 'sendmail' are supported as a different configuration.
 * For 'smtp', the SMTP server's settings must be set, while 'sendmail' requires no configuration.
 *
 * @global string $CONFIG->emailer_transport
 */
//$CONFIG->emailer_transport = 'sendmail';

/**
 * Configure emailer SMTP settings
 *
 * This setting is only necessary if the above emailer transport is set to 'smtp'.
 * Please refer to https://docs.zendframework.com/zend-mail/transport/smtp-options/#configuration-options
 * and https://docs.zendframework.com/zend-mail/transport/smtp-authentication/#examples
 */
//$CONFIG->emailer_smtp_settings = array(
//    'name'              => 'localhost.localdomain',
//    'host'              => '127.0.0.1',
//    'port'              => 25,
//    'connection_class'  => 'login',
//    'connection_config' => [
//        'username' => 'user',
//        'password' => 'pass',
//        'ssl'      => '', // OPTIONAL (tls or ssl)
//        'port'     => '', // OPTIONAL (Non-SSL default 25, SSL default 465, TLS default 587)
//        'use_complete_quit' => '', // OPTIONAL
//    ],
//);

You would have to uncomment the config options. $CONFIG->emailer_transport would have to be smtp. And the array entries to be set in $CONFIG->emailer_smtp_settings depend on the smtp server you intent to use.

Thank you ever, ever so much, this is exactly what I was looking for!

Exactly the hint I was looking for, I certainly do thank you for your attention in answering this post.

@Claudius: please update if its working for you. For me it was not working even after making those changes. 

Please see Configuring SMTP settings : Elgg.org