My baby was down with cough and phlegm last month. I learn from this blog post about steaming an onion and get its juice to relieve cough and phlegm.
Get one of those big red onions (not shallots) and cut them into eight quarters. Steam for about 10 minutes. You should be able to get around 20ml of onion juice.
It's recommended to give only one teaspoon (5ml) of onion juice to 6-month old babies. This remedy might not be suitable for young babies below six months.
I only gave one teaspoon to my baby and he didn't really like it, hahaha. You might want to give this remedy a try if your baby has the early stages of cough and phlegm.