If you are a cigarette smoker, you will probably have to quit smoking for a month or more to allow your lung tissue, specifically your inflamed bronchial tubes, to heal.
The second possibility is that, even though you may not have ever had the diagnosis of asthma, the viral infection (cold) you had may have been severe and irritating enough to result in asthma. This is not a rare scenario and, when it happens, cough rather than wheezing may be the main symptom. If that is the case, you will probably have to have asthma medication for the cough to be controlled. If things play-out this way, it doesn’t necessarily mean that you will continue to have asthma indefinitely.
The third possibility is that your “cold” was actually whooping cough (pertussis). That type of cough can linger for 3-6 months.
If you prove not to have asthma, aren’t smoking and didn’t have whooping cough,you should contact your doctor and request a prescription for a strong cough suppressant You could try an over-the-counter cough suppressant with “DM” and if that doesn’t provide relief of the cough, then call or visit your doctor to make sure that nothing more serious is causing your cough (you may need a chest X-ray to do that) and get a stronger cough preparation such as Tussionex.
Good luck
I would get a throat culture from your doc- it could be bronchitis, or just a virus but either way, get to a doctor...good luck!