Hope both you and your little one are doing well. Home remedies are best at this age as babies are prone to cold and cough due to season changes. I have been always using home remedies for my baby. Heat mustard oil with 5-6 garlic cloves till it turns brown. Then cool down the oil till mild warm. Make sure that it is not much warm, put it on your elbow to check it. Then massage the oil on the chest and back, on the feet and palms well. If the weather is not very hot then put on socks. But keep an eye, if the baby is uncomfortable then open the socks. This works very well.

Hope this helps.