On the phones without a mailbox configured, do you intend to set a different voicemail profile with a unique voicemail pilot? I ask this because if you don't, the steps mentioned below will effectively replace your Opening Greeting call handler. If you choose a different voicemail pilot you could simply set this extension on the call handler, leaving your opening greeting intact.

You need to create a System Call Handler (not a directory handler). Set your greeting as desired and choose the After-Greeting action to transfer the call to an alternate contact number. Specify the extension you want and choose release to switch.

If you did not choose a different voicemail profile/pilot number: You will need to supercede (or replace) the Direct Call Routing Rule table entry of the Opening Greeting. The table is an ordered list so add your new call handler to the table AFTER the Attempt Signin and any other conditions you are trying to match.