To a 3-month-old baby who was coughing heavily, the pediatrician presc…

To a 3-month-old baby who was coughing heavily, the pediatrician prescribed antibiotics and syrup. They took a week of medication, now it seems to be improving but he still coughs sometimes with phlegm. I'm doing inhalation to finish treating the cough completely. Should I continue with inhalation or what can I give to make the cough go away for good?