In section II, there is alliteration of the hard "c" sound that begins the words "comes" and "consciousness" (II.1), the "s" sound that begins "stale smells" (II.2), the "s" sound the begins "sawdust" and "street" (these two words are also an example of consonance, with their final "t" sound). There's actually lots of consonance from line to line in this section on the final "s" sound: "consciousness" (II.1), "smells" (II.2), "press" (II.4), "stands" (II.5), "masquerades" (II.6), "resumes (II.7), "thinks" and "hands" (II.8), "shades" (II.9), and "rooms" (II.10). There is assonance on the long "a" sound in "faint stale" (II.2), the long "e" sound in "muddy feet" (II.4), and the long "a" in "raising" and "shades" (II.9).
I think it is reasonable to suggest that the poem takes a sex worker or prostitute as its subject. The first section talks about being outside; it's cold and unappealing and turns to night as people light the lamps (prostitutes are often associated with night, even sometimes called "ladies of the night"). Section II jumps to the morning, the smell of beer in a "thousand furnished rooms" (like cheap hotel rooms people might share with prostitutes). Section III describes the person waking up, waiting in bed, and a discussion of "The thousand sordid images / Of which [her] soul was constituted." Some might argue that this would describe someone in this line of work. I like your idea, however, of interpreting the subject as any person in the modern era: someone who lacks real connection with others, someone who, perhaps, feels "sordid" and dirty and worthless, some "infinitely gentle / Infinitely suffering thing." There's no reason one's interpretation must include the sex worker angle; the narrator does not specify a particular person. Your interpretation, I think, opens up the poem to many new avenues for discussion of modern life and identity.