J.M.
His cough could be a result of some sinus drainage. Most commonly when children have a head cold they can get an ear infection. Since he is on antibiotics, it could be breaking up the cold and mucus and causing some drainage in the back of his throat. Generally, you should have a follow up appointment for the ear infection, and the doctor should look at him them. But, if this persists for more than a week after the antibiotics are done, I would take him back to the doctor. (Sometimes if you just call the doctor they can give advise over the phone and save a trip to the office)
Best of luck,
J.