After A Week Of Being Treated For Upper Respiratory Viral

After a week of being treated for upper respiratory viral infection with no production of phlegm despite wheezing in soft palate and neck area , doctor started me on Methylprednisolone from decreasing pack yesterday. Since it was just after dinner, took all 6 pills at once with ok of two pharmacists. Began to work in an hour. Closed ears opened up etc. Now feels like a runny cold.
Today, I took one pill before breakfast and may take remaining 4 together after dinner. I am coughing and wheezing from trachea area.

Question : what can I take along with the decreasing steroid pack to break up the phlegm thereby stopping the coughing. I want the most mild thing since I am on blood pressure medicine and quite a few other pills. May I take an expectorant with the steroid pack ? I do not want to take a cough suppressant. Only guaifensen 100 mg

Thanks
Ann