Note that a high voltage will lead to side reactions, such as reduction or oxidation of you're solvent. A high potential will than give you large currents, but if those electrons are used to form hydrogen and not copper or zinc it isn't much use.
I should have used the term current density instead of current in my earlier post. If you looked up the Butler Volmer equation, you might have noticed the A in there. Increasing overpotential leads to a higher current density and thus to a higher current. But, using a larger electrode at the same potential also increases current since the current equals current density times surface area.
That's why I pointed out that it is good to use large electrodes, since then you can maintain high currents at a relatively low current density. This way you don't need as high overpotential, which reduces problems with side reactions.
Good luck with the experiments.