The story
From mossy + back (in sense 2 (“person who stayed hidden to evade conscription during the American Civil War”) probably a variant of mossback, which is attested earlier), alluding to someone or something that is so old or has remained so unmoving that it is as if moss has grown on them or it. Sense 3.2 (“menhaden”) may be influenced by mossbunker.
Etymology from Wiktionary, CC BY-SA 4.0.