The simplest way is to turn exim on the old server before yo do the backup. This works well if you can complete the move within four hours.
You could also turn on DNS for all users on the old server, and then log into each account manually and set new mx records and uncheck the Use this server option (all under MX records).
Or do the same manually, removing all domain names from the /etc/virtual/domains file on the old server, and changing the MX records in all the zone files in /var/named using the bulk file editor of your choice.
If you decide to leave exim turned on and use the second method (either manually or through the control panel), be sure that the new server already has the domains set up and the email users and forwards as well.
In other words, we prefer the first method.
Or as one final method you could set up a backup mailserver. How to do that is well beyond the scope of a response in these forums; whole websites have been devoted to setting up backup mailservers.
Jeff