I've been pondering on answering your question, I am not sure however if I can make the answer simple and how it affects the ISPConfig on your server (although I'm guessing it won't affect ISPConfig).
So here is a brief outline....
Configure your second IP to your NIC interface ie: eth0:0
Give the IP a name (FQDN) and make sure that name is added to the second IP both in your dns and in your /etc/hosts file
Change /etc/mailname to this new FQDN for the second IP
In /etc/postfix/main.cf you'll need to alter a few things for postfix.
inet_interfaces = 127.0.0.1, your.second.ip.address
(note if you also have a private lan attached on another network card ie: eth1 - you'll need to also add that network)
mydestination = "fqdn of your second ip", localhost, localhost.localdomain
(note you should perhaps leave the existing name of your server that is in here as well - as you most probably are using local mail delivery for that to)
You will also note the "myorigin = /etc/mailname" is in here as well - that's why you had to change that mailname file.
Now, you will most probably have to create a new ssl cert for postfix under the new name - if you created one for the old name.
restart postfix, a reload won't do when you change the interfaces.
Thats basically it.
Things to watch out for: try and get a reverse dns entry for your second IP to match the fqdn you gave it - your provider will most probably need to do this.
Questions? fire away...