Outgoing Email Settings

There are multiple Places to update Outgoing Email Settings on SharePoint Server.

1. System Settings >> Configure outgoing e-mail settings.

2. Application Management >> Manage Web Applications, select the Web Applications.
    Drop down - General Settings >> Outgoing E-mail

Make sure the Settings were same across all the Web Applications.

You can also specify different From address and Reply-to address based on the purpose of the Web Application.

Also you need to work with your Network Admin to make sure the Static routes are configured correctly on the SharePoint Front End/App Server.