For a baby of 3 months with a persistent cough, it is not recommended to give syrup due to the associated risks and limited benefits they provide. Instead, hydration is key to soothing the cough. Make sure your baby drinks plenty of fluids such as milk and water. If your baby continues to cough for more than 10 days, has a fever, or difficulty breathing, it is best to consult a doctor. Keeping the environment humid with a humidifier may also help alleviate the symptoms.