Thanks for the information!
The cough you report can indeed contribute to chest discomfort, as it is a symptom that involves a lot of muscles and joints
For now, the best treatment will be non-pharmacological: I suggest nasal hygiene with saline solution, like Hydrasense, Sinus Rinse or others. By cleaning the sinuses from the "front", we prevent the discharge in the throat and we can greatly reduce the cough. This can be done as often as desired during the day, there is no maximum. It's a bit unpleasant but very effective!
You can also apply a Vicks type ointment on the chest or back, if needed
However, if you notice that the situation persists beyond a week, or gets worse, or is accompanied by other symptoms, you should consult a doctor again.
I hope that I have been helpful to you and I am available if you have any further questions!
Tom Samaha, pharmacist