You could ask every customer that calls how they found you but that's not a great solution either because it's a)not good user experience and b) it's not time-efficient.
Sounds like an organizational issue to me. Head of call center shouldn't have ANY issue reporting how many calls were made to a given number. If that's the case, you need to hire a more competent person for the job. You need to convince your superiors the importance of tracking, how it affects everything from budgets to strategy. Get to work! :)