My son has a lingering chesty cough after every virus he gets!
We used to use baby vapour rub may upgrade to Vicks 👍🏻
I did the vicks on feet trick recently too.
Honestly i would avoid antibiotics and let them ride it out naturally whenever possible.
Most things are viral which anti-biotics do nothing for anyway.
My friend says the Calpol plug ins are good for their little boy when hes congested as well.
Vicks vapour rub on feet with socks on! Works wonders for coughs :) I’ve used this on all mine when they’ve had colds x