Cat vomiting and coughing regularly

My cat has been coughing more than once per day. He seemed as if he wanted to cough out a hairball but isn't able to do so. We also suspected that it could be asthma. Recently, we brought him to a vet for an x-ray and ran blood tests as well but nothing seemed to be particularly alarming. The vet reported that my cat had some form of viral infection which could have caused the coughs. We were given antibiotics and medicine to counter the infection. After the visit to the vet, he has been coughing more frequently and vomiting as well. Initially, we assumed he managed to successfully cough out his hairball but instead he has been vomiting out his food. His behaviour is normal. He still plays, eats and poop regularly. I don't know what's causing his coughing and vomiting and it really worries me. It's been 2 days since the last vet visit so I believe it would be better to wait at least a week before visiting the vet again, unless it gets worse of course.

It would be great if you guys have any ideas what's causing all this.

Thank you.

I would just take him back to the vet for further checks.

What kind of medicine did your vet prescribe? Vomiting could be a side effect and if it's disrupting your cat's ability to eat, you might want to call your vet for an alternative type of medication. 

My cat is currently on prednisolone and clavaseptin for a suspected URI infection as well (he was once coughing everyday for about two weeks). On the third morning, he vomited a bunch and didn't want to eat for the whole morning/afternoon. Then after that, he started eating again and no vomiting. I suspected it was the antibiotics kicking in making him vomit and act lethargic.