Is anyone a lactation consultant?

Hi! I have a phone call into my lactation consultant, but thought maybe I could get help here. My 16 month old (not breastfed anymore) has a nasty cold/cough. I'm breastfeeding my newborn and from what I know I should be providing immunity to the newborn through my breast milk. So my question is would it be helpful at all to give my 16 month some pumped milk? I know it can't hurt but I don't want to waste it if it likely won't help him get over it faster.