G'day Alexa,
Kennel cough is basically a doggie cold, and like us it can be a minor irritation or a huge hassle. Monitor him carefully, being underweight and also young when you got him, I'd be more inclined to go to a vet, mainly because he wasn't in the best of health and on the mend in the first place. Your vet should have told you his ideal weight by now I'm guessing, so that should also be monitored carefully. (enjoying the cuddles too I bet )