Last May does indeed mean the May of the year before. Unless however it's June or later and you're talking about both the May of the year you are currently in AND the subsequent May.

If it's near the end of the year or early in the next year and you say "this may" it means next May. But if you say "this may" earlier than that, it will be assumed it is the May of the year you are currently in.

English is a superb language. Especially with cultural thingies thrown in too.