If you need to send lots of e-mail addresses via your Exim MTA, you will most probably want to do so from a new IP address, so you can preserve your main one. Configuring this is as easy as 1-2-3 :) Open exim.conf for editing with your preferred text editor and find the following lines:
remote_smtp: driver = smtp
After them add the following one:
interface = NEW_IP_ADDRESS
and restart exim. This will force it to send from the specified address.
NB: This IP address must be configured to one of your network adapters beforehand.
Disclaimer: This will not work on a shared hosting environment. The purpose of this article is to help someone, who has just started administering a hosting/mail server.