An adult allergy or pulmonary consultation would be appropriate. Also such specialists sometimes identify themselves as asthmatologists. Some academic institutions such as National Jewish Health in Denver, Colorado, the University of Pittsburgh Hospitals in Pennsylvania and Duke University in Durham, North Carolina offer highly specialized, expert asthma consultation. Your description of the episodes between October and May does suggest the diagnosis of seasonal asthma. Factors to be considered during that time frame, in addition to viral infections, would be cold air, increased air pollution and indoor allergens, the concentrations of the latter that increase as houses are closed-up during the winter.
You should definitely get a flu shot and, to the extent possible, avoid crowds and especially individuals with respiratory infection symptoms, during the winter months.
Given that your problems begin in October, you and your doctor might want to consider starting your asthma medicine regimen in September.
I used a pulmanologist to finally get a diagnosis. There are allergist who treat asthma also. I am currently using one. I used a pulmanologist to get diagnosed because they could do a methocoline challenge test in there office and I didn't have to go all over the city getting tests done.