Sounds like he might have kennel cough if that is all he is doing. I had a pup with it once and it is horrible to hear them cough like that. Does he have a fever? That is always a good thing to keep track of to tell you how things are going. I can't remember if there was anything the vet gave me to help. It seems that it just has to run the course. You could make him an herbal tea to drink if you had some. He might not want to take it though. You could warm him some chicken broth or some kind of broth, dilute it with water so it is not so salty and give him that warm to drink, that would help sooth him some. Just repeat at times over the day. I would suggest adding some honey, but not sure if that is o.k. with a puppy. Might want to goggle that and see if it comes up. It really is sad how the vet world is now too busy due to a holiday to help a poor little pup, but that is how they've become. Good luck getting him over this. We will keep our paws crossed for you.
Nano, Mateo & Talia