Enormous in small spurts. Your word and recommendation go far. You can "make someone" to the extent that they showed up to the party and didn’t order long noodled spaghetti for lunch with the CEO. The presumption is that you know the employee you are pitching to the company vastly better than the company does.
By sending out the occasional "refresher list" email to everyone in your rolodex, you have the power to remind execs that there's a potentially more satisfying job out there somewhere, and one more snide comment from the CEO at the water cooler might be enough to get them to hit "Reply"….