It is common for a baby to experience coughing after having a cold. This could be the last symptom to go away and may persist for a few weeks in some cases. It is important to practice good nasal hygiene to help with the coughing. Even if there are not many secretions, a small amount can still trigger coughing. You should see gradual improvement each day, but if the cough persists for more than a month or gets worse, it is advisable to consult a doctor as a precaution. Hope this information helps to reassure you. If you have any further questions, feel free to ask.