TBH, if it was my kid, and they only had a cough, I'd wait it out for a day and see what happened. Maybe give a little tylenol if he seems uncomfortable. But...I AM NOT A DOCTOR. My advice means nothing.
With only a day of it I'd just monitor it. Set up a humidifier in his room at night. If he starts wheezing or can't catch his breath or if he develops a fever, definitely call the doctor. The wheezing is indicative of croup and there's always a chance of it being bronchitis or pneumonia. Chances are, it's just a virus and will run its course and is more annoying than anything (says the person whose entire house has been hacking up lungs since the beginning of August and every one of us has been to the doctor for it). Hope he feels better!
Re: Phlegmy Cough