No, there is no special name for the closest supermoon.
Besides, the very name “supermoon” was coined by an astrologer, NOT an astronomer, and it is basically all just media hype. During a so-called “supermoon,” our satellite is on average 15% larger and 5% brighter than during a regular full moon. Since the previous full moon dates from a month before (the etymology of “month” is the same as that of “moon”), it is impossible to remember precisely how bright and big the lunar disk then was. So anyone and everyone who claims that they noticed the moonMoon as “bigger and brighter than usual” during a “supermoon” is actually just being influenced by the media.
Astronomers see nothing special with a “supermoon”—the only “advantage” I see is that the media then talk more about astronomy, so it’s some “good and free advertisement.”