[link to original on tumblr]
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I'm pretty sure that I've previously noted the descending chromatic phrase in the backing vocals during the choruses. After reading some of the Wikipedia article on lament bass, I'm not sure if it perfectly applies here, but as a chromatic descending phrase, it does provide an impression of sadness, which gives a melancholic edge to the lyrics. That descending chromatic figure questions the hope in "Maybe after he's gone / She'll come back / Love me again."
I'm fairly sure that the verses are in E minor and the choruses in A major. So that descending chromatic figure works against the lyrics, but also against the relative happiness of A major (compared to E minor, which - as a minor key - is usually perceived as sad).
Yet there are also an ascending chromatic phrase in the piano part during the choruses (after "Love me again"), which I don't know how to reconcile. Maybe there's a conflict between optimism (characterized by that ascending chromatic phrase on piano) and pessimism (characterized by the descending chromatic phrase in the backing vocals).
Incidentally, those ascending chromatic phrases are almost identical to those in Argent's "Christmas for the Free." "Maybe after He's Gone" has A B C C#, and "Christmas for the Free" has A B C C# D (of course, the A isn't part of the chromaticism, but it's in both phrases). The rhythms are different, but the notes are the same.