Chronic cough? 5 month old

Hello everybody! FTM HERE and My son is 5 months old and since he was 3 months old he has a cough. I told him in to his doctor and they gave me an inhaler and an allergy medicine. The medicine wasn’t doing anything so I stopped using it. The Inhaler does help him a lot. Well these pst couple of days this cough has gotten worse. Stuffy nose, cough attacks, breathing with him mouth open, phlegm that comes back when we remove it. I clean out his boogers and it doesn’t help. At this point I’m not sure what to do anymore. He doesn’t have a fever and acts himself, would it be a good idea to take him to the hospital so they can check him out? I’ve taken him to multiple appointments and they just say it’s going to get better. He had a really bad cough attack this morning and he couldn’t breath and started crying loudly after it was over. Any advice? Please