I would try benadryl first to dry up the cough. Check with the nurse at your pediatrician's office first (if you call them they can OK this over the phone). This is usually what the recommend for my kids with a nagging cough - it's probably post-nasal drip. Once you dry up whatever is still lingering in the sinuses, the cough will stop. If that didn't work, then I would bring him in after whatever period your doctor's office recommends. At my pediatrician, it's a couple of weeks of just coughing (no other symptoms) before they say bring the child in. With the common cold, the cough usually lingers for 10 days.
ETA: just read that it's been less than a week - definitely just let it work its way out. After a couple of weeks, that's when we try something like Benadryl to get rid of the last remnants that are hanging around. But right now, it's the body's way of expelling the virus.