There is a common over the counter mix of guaifenesin (expectorant?) and dextromorphan (cough suppressant?) that I find helpful. I believe it's called Mucinex DM, but I always get the generic. The one reduces the urge to cough, the other makes coughing more, shall we say, productive. Both drugs are weak enough that I don't notice any side effects at all. In fact the benefits may be entirely psychosomatic. Good enough for me!
My wife mentioned this today, but until your post I hadn't heard it used by a cyclist, or better yet, a long distance cyclist. When in your ride do you take this? When you are done for the day or before?