Prolonged Coughing. Causing Sleeplessness. Recommendation For Tests?

Back in January I had bronchitis + flu with a bad cough. Ever since then, my immune system has been pretty weak (as I caught another flu earlier this month. No bronchitis this time, but sinus congestion). It has all cleared up, but my cough from January has lingered to today. I cough less now that the weather is warmer, but I noticed that I cough mostly during the nighttime when I am lying down on bed and when I feel cold air. I have had my annual checkup with my family doctor, but she did not address my coughing much. Just listened to my lungs and said it was OK. However, this cough is very annoying as I cannot get much sleep on nights when I cough. Who should I go to and what tests should I get done to get this checked out? Should I consult an ENT or pulmonologist? Thanks..