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If you have to configure Exim mail server to act as RELAY Forwarding SMTP through another mail server and nomatter, you set up in dpkg-reconfigure exim4-config command a hostname still connecting to localhost port 25 shows a hostname identical as configured hostname for server. You will have to explicitly instruct exim to desired hostname like so:
debian:~# cp -rpf /etc/exim4/exim4.conf.template /etc/exim4/exim4.conf
Then edit /etc/exim4/exim4.conf
Add somewhere in beginning of config file
primary_hostname = DESIRED_HOSTNAME.COM
To make exim load new settings, restart SMTP:
debian:~# /etc/init.d/exim4 restart
Stopping MTA for restart: exim4_listener.
Restarting MTA: exim4.
debian:~# telnet localhost 25
Connected to localhost.
Escape character is '^]'.
220 DESIRED_HOSTNAME.COM ESMTP Exim 4.72 Tue, 19 Mar 2013 19:02:52 +0000