Steroids for a cough? That's pretty extreme treatment and results in some very bad side effects. Maybe all you need is a mucolytic (mucus thinner). Large doses of vitamin B5 work well as does over-the-counter guaifenesin. I use acetycysteine by nebulizer. Also, drink plenty of water.
By the way, the mucus might be caused by an overproduction of Candida albicans (the same yeast/fungus that causes thrush). It is a very common problem that is made worse by steroids. Candida loves steroids and antibiotics and sugar. Look it up on the internet.
For years I suffered with cough and mucus, and asthma as well in the past. In 2005 I was sick for two months, consumed two rounds of antibiotics, nothing worked, and I was misreable, unable to sleep. X-rays did not show any problems. One day a relative of mine took me to a homeopath. One simple dose, believe it or not, same night I slept like a baby. Try homeopathy, you got nothing to loose. If it works, you will see an improvement in a couple of days. In my case, the relief came exceptionaly quick. If you do not believe in homeopathy (I never did until then), try at least Vitamin C 500mg every day.
I have been involved in alternative medicine since before it was fashionable. I am happy to see a fellow adherent. It saved my daughter from being a cripple, and it diagnosed some chronic ailments that were undiagnosable by traditional medicine. Cheers, and spread the word.