Baby cough

Ok so my baby got a chest cold and was super snotty and coughy just nasty congested ghee would cough till he threw up from all the snot. Well he got Over it and went straight into teething so is still super snotty and just can't get over that cough and throwing up from all the snot. We have a diffuser with breath in it a humidifier warm baths with breath baby vapor rub I don't know what else to do