To my mostly AE ear, there is a slight difference between "at hand" and "on hand". I might say "I like to have flour, eggs and sugar on hand in case company comes and I need to bake a cake." Here "on hand" means in stock, available. However, I would say, "When I'm burning trash, I always have a source of water at hand, in case it gets out of hand." Here "at hand" means nearby, within reach. "Out of hand", which I give you for free, means out of control.