Transfer/Renewal of expired H1B.

I got my H1B stamped from Organization A,but did not get a chance to travel. Now I am in organization B and have my visa as expired. Can any other organization get it renewed/transferred and if yes can it be any time in a year or I’ve to wait till filling period i.e… April

As I am having expired H1B, I am cap exempted so can renewal be done at any time in an year or it will be done at the time of fresh fillings only?