Here in the gloomOf my lonely roomI hold her handkerchiefAnd smell her sweet perfume
In the first verse, the narrator explains that he's alone. Here, he "hold[s] her handkerchief" almost as a substitute for holding her hand. The two phrases start the same way.
The lines "I hold her handkerchief / And smell her sweet perfume" exhibit alliteration, and the euphony of this effect gives some indication of the pleasantness of either the situation the narrator imagines or his actual memories of the past.
The initial sounds of this pleasant alliteration contrast with the line-ending rhymes in the previous couplet, where the narrator's sadness and isolation are highlighted. Their opposite positions emphasize this difference.
In the first verse, there's a slant rhyme between "alone" and "home" ("It's so bad to be alone / Oh, baby, come home"), and this imperfection indicates the narrator's lack of sufficient companionship.