There are dual voltage EVSEs (I use one called Duosida), but if the label on the back of yours only lists one voltage, it won't work.
And keep in mind that depending on which 240v receptacle you have wired up, there's often no neutral. On a dual voltage EVSE, the neutral wire becomes the second hot lead, no neutral, just ground. Though I suppose that only matters if you're trying to step down, not step up.