So rather than “leave a message” you might offer to send a gatekeeper a one-page pdf or email that explains what you do, the companies that have chosen you, the problems you solve with representative results. You offer to send that (they don’t have to pretend they are taking a message) and then call the gatekeeper back in a few days to see if Mr. or Ms. Big wishes to meet. That way you don’t have to keep calling back.
