The rash does not seems to be associated with the previous disease as it took weeks for the rash to develop and as you described it.
You must consult an ENT specialist
There is cold followed by cough there could be chest infection
You must get done chest xray, CBC ,ESR to rule out the source of infection.
YOu also need to get examined for any infection in nose and sinuses
You may require CT scan nose and paranasal sinuses
If in doubt do reply