Why wouldn't you have ONE mail server (software) running multiple domains?
For example... you can set up a single mail server to handle email for:
...without the need for having multiple servers/software running.
That being said... it IS possible... you just have to keep an eye on your IP addresses and the ports you're using.
In the end its WAY easier to manage if its just one server and multiple domains...
Hope this helps.
In regards (specifically) to migrating an old mail server to a new mail server...
Its always nice to have TWO static IP addresses. You can then create DNS entries (with a short TTL) to point to mail2.example.com
Test out the server all you like... when you're happy with it change the DNS to make it your new "mail.example.com" and then create a record for "oldmail.example.com" to point at the old mail server.
This also helps if you need to migrate user mailboxes, etc over or some user in a panic needs to (somehow) get back to the old mail server to get something.