Symptoms
By default, port 465 is used for TLS connections while port 587 uses STARTTLS. But there may be a situation where, when using a third party client, port 587 is required to use TLS immediately upon connection.
Workaround
According to Exim's documentation, the TLS-on-connect may be added to additional ports by way of the following setting in the Exim configuration:
tls_on_connect_ports = 465
To access the setting, navigate to the Exim Configuration Manager from WHM: Home » Service Configuration » Exim Configuration Manager. From there, click on the Advanced Editor tab and then search for "tls_on_connect_ports".
From there, additional ports may be added as necessary by appending a semicolon and then the port number like so:
tls_on_connect_ports = 465:587
Refer to Exim's documention regarding their default configuration for additional information.