It's not perfect and the user needs to remember to set the From address, but it works. When the user replies, he can select the DG as the From address. When messages are received by the distribution group, they are delivered the users mailbox, just as they are now. The administrator needs to give the user's primary account Send as permission to the address of the distribution group. With the second SMTP address set up as a distribution group address and the primary address as the only member, mail to the DG is delivered to the users mailbox. Next to separate mailboxes, using a distribution group (DG) may be the second best option. It will automatically choose the sending account based on the folders you are viewing when you begin a new message. None of the methods are perfect, although Outlook 2010 (and up) handles it the best, by allowing two Exchange accounts in the profile. You can use shared mailboxes instead (does not require an Office 365 license) but you will need to select the shared mailbox address when you send new messages.
MAC UTILITY FOR TURNING EMAIL SIGNATURE INTO CONTACT INSTALL
(Remove the secondary addresses from the mailbox before setting up the groups.) Give Send as permission to each distribution group.