You must correctly set up the SMTP "sending" server name in the email program. This sending SMTP email server name must match the name provided by the ISP that the router is using to connect to the Internet.
Internet Service Providers (ISPs) tend to block access to all SMTP server that are not part of their own network. Therefore, if you are using a laptop on several different networks, you would probably need to adjust the sending SMTP server for each.