If you have a cold and begin feeling better but start coughing then yes that coughing can be the last stage of a cold and is usually a good sign.
Some people who have the cold virus even cough of phlegm which can be yellow or clear and when the phlegm is getting clear it means that the cold is going away.
A cold virus usually lasts between 1 week to 10 days but sometimes can last a bit longer.
Usually though after a few days to a week you begin feeling a bit better from the cold virus which is a good thing so if it's been a few days or a week and you're coughing then it is usually a sign that the cold virus is leaving your body.
The coughing can last a few more weeks to a month depending on how bad your cold virus is.