No, there is no special name for the closest supermoon.
Besides, the very name “supermoon” was coined by an astrologer, NOT an astronomer, and 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 Moon as “bigger and brighter than usual” during a “supermoon” is actually just being influenced by medias.
Astronomers see nothing special with a “supermoon”—the only “advantage” I see is that medias then talk more about astronomy so it’s some “good and free advertisement.”