Coughing for 3 years now when fall/winter comes

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So when my son turned 3 (almost 7 now) around fall and winter my son gets this cough. His doctor doesn’t seemed worried he either gives him a steroid cough medicine or allergy medicine. He also says it’s seasonal allergies but idk in my heart and mind I think it’s maybe more. My baby wakes up coughing it’s dry or sometimes loose. I give him water or orange juice and it usually helps. The medicine he’s taking I can only give to to him in the evening (when he comes home from school). By noon he’s not coughing that much just a cough here and there (I know this cause I have lunch with him 3x times a week or more and I also ask his teacher). Morning time is when he’s coughing the most. Well I’m starting to think this maybe asthma and I should take him to a specialist. My son does have allergies and it hits him in March through May. What should I do listen to his doctor or seek a allergy/asthma specialist?