You can add in multiple email addresses for contacts. Emails from all addresses will then show on the email tab, so you can leave old email addresses on their record to continue to see emails to and from that address. To do this:

1) Open the client record in question

2) Go to the 'Contacts' tab

3) Click the blue 'Edit' button

4) Under the contact's email address, you should see the option to 'Add email address'

5) Put in the new email address

(Note: whichever email address is listed first will be the default address for emails to that contact)

6) click the blue 'Save' button when you're done.

From now on, by default, all email sent to that client and contact will go to whichever email is listed first in the list of email addresses for the contact.