That's good. Very good.
Sometimes, if it's KC, it can take a long, long time for the lingering cough to go away, sometimes weeks, It's just like when people get colds and have a cough that hangs in there after they're feeling better.
I love and fully support shelters. But they are a breeding ground for virus and bacteria, since they have so many different pets coming through all the time and are often short-staffed so cleaning thoroughly isn't always possible. He could have very easily picked up KC while he was there. The same thing happened with my kitten Gracie. I was originally fostering her for the SPCA, but decided to adopt her. She was perfectly healthy the entire month I had her, but one day's stay at the shelter to be spayed, and she picked up a respiratory infection.
Your vet may be able to suggest or prescribe something to help with the cough. Antibiotics are more to help with a secondary bacterial infection, if one happens, not the Kennel Cough virus. But not all antibiotics help all infections, so if he gets worse or doesn't improve, check with your vet. They may prescribe a different antibiotic.
He's an adorable dog.
