My baby had a phlegmy cough, which developed into Bronchitis, and breathing problems. When I mentioned to the Dr. I'd been using a Vaporizor, he told me not to use it, as it worsened the problem. Vaporizers and humidifiers are good for dry coughs, especially in dry weather. Robitussin with the guanif. ingredient is good, and Mucinex works well, too, but I don't know if it's OK for a child as young as three. Try to keep your child upright when not sleeping, as laying down makes it feel worse, although a child has to sleep - elevating the head could work.
However, he did say a hot shower/bathroom steam in a clean bathroom would be good due to the limited time. And yes, antibiotics may be needed. The earlier, the better, once Dr. hears rattling.