inhaler not totally effective?

I'm a cough-y asthmatic rather than wheezy and a thought struck me today

Should the inhaler fully settle the cough?

A couple of times this week the asthma cough has been triggered (once in my dancing lesson) so i've taken the inaler and it's fully settled it within a few minutes. EVERY time i go running i end up coughing for at least a few hours afterwards, i've thought it was normal (well it is for me) but the effectiveness of the inhaler when i've taken it at the first struggle just made me consider whether i should maybe think about seeing can i get my medication altered so that it's better for my running