When poets refer to other great works, people, and events, it’s usually not accidental. Put on your super-sleuth hat and figure out why.
Later on in the Exeter Book there's another poem called "The Husbands Message." This poem is written from the perspective from a husband to his wife or betrothed, who is far away in another land. It deals with the same themes of sorrow and isolation present in "The Wife's Lament." Could "The Husband's Message" be a response to "The Wife's Lament"? There is no conclusive evidence to link the two together, so we're limited only to speculation. But our imaginations are running wild.
Also, a not uncommon opinion amongst scholars is that the poem is, in fact, one giant shout out. We can't fully make sense of the poem, the theory goes, because it is part of, or at least makes reference to, another narrative that would have been familiar to contemporary readers. Ambiguous passages, like the part about the husband's kinsmen (11), or the reason why the speaker was exiled in the first place (5, 15, 28), would be clear within the context of this other story, and thus wouldn't need to be explained in depth.
So while the story seems pretty dang unclear to us modern readers, contemporary readers would fill in the gaps based on their knowledge of the other narrative. We guess we just missed the punchline.