A better solution would be to create 2 profiles in their email reader for the same address, then use the mail server IP address for the old host in the incoming and outgoing server definitions and then in the second profile do the same thing with the IP address that you receive here that way the email program is hard coded to check the exact IP for mail on both hosts.

once propagation is complete they an delete the profile for their old host and update the new profile to use mail.domain.com instead of the Ip for incoming and outgoing mailservers.

This allows you to check email on both hosts during propagation. A little more advanced but it works well. You'll have to ask the other host what the IP is or ping the domain that you have listed in there now and then when the ping returns an IP plugging in the IP instead of the domain should work fine for you.