Support Case: How to Send an "Outgoing From Support Rep Message" to Multiple People

When a support rep sends a message to a customer from within a case using the Outgoing From Support Rep Message, an email with that message gets sent to the customer contact attached to the case.  However, I'd like to add multiple people to that email so that any outgoing message goes to the entire group. 

If you're familiar with Jira and the Watcher concept, this is the same thing.

I've created a custom record type called Additional Customer Contact and attached it to a subtab on the case.  This allows a user to add more than one customer contact to the case.  I've created a saved search that is triggered when anything is updated on the case and sends an email to the case contact plus all the additional customer contacts.  This works great, however it's not picking up that an outgoing message has been sent.