G.♣.
If you took him to the clinic 4 days ago, I wouldn't look at the cough as being from a cold 12 days ago. If he was still feeling sick 4 days ago it's perfectly normal for him to have a cough.
I usually have a cough for at least a week after a cold. It's usually an irritating dry cough and not usually with chest congestion.
If he has a lot of congestion or isn't feeling better, it's worth a call to his doctor in the morning. Otherwise it seems safe to say he is on the road to recovery.
I probably wouldn't fill the prescription without talking to his doctor first.