@maxdancona,
I used to go to a doctor only when my situation got worse, but a few years ago, I went to a good ENT and said that I wanted her to get to the bottom of my condition no matter how many tests had to be run, and that I should see her periodically, regardless of whether I had a cough or not.
I have kept going back to her and now she trusts me with enough codeine to control my situation and prednisone when it really flares up. I made one bottle of cough syrup with one refill last for 11 months. I would probably have made a full year, but for catching influenza A. My ENT understands exactly what I have and acts accordingly. I can tell you, though, that numerous tests, including CAT scans which I more or less demanded, have revealed very little wrong. As I said, her thinking is that it's multifactorial, but involves, cough variant asthma. Once or twice when it was really bad, they gave me an injection of steroid, which helped a lot.
Like you, I do get a very bad cough at the end of an illness, but I often get it without the illness too, if something makes me cough a a few times.