☆.H.
My 5 year old has gone through 2 of those types of colds since school started. Both lasted 6-8 weeks! In addition to antibiotics he was also given an inhaler. The doctor explained it's not necessarily asthma but that it's normal every few years or so for a child to get a cold where the phlegm gets too far down for them to cough it all up. Also, we have been using an antihistamine to dry up his nose (the decongestant used in kids cold medicines these days is not very strong or effective)
So, anyway, there may be more that can be done. It's worth the appointment.