Lyric discussion by offtehdeepend 

Cover art for Film Noir lyrics by Gaslight Anthem, The

The girl's name, Matilda, is probably a reference to "Waltzing Matilda," an old Australian song about a homeless man. The reference ties into the image of him being alone and out on his own with nowhere to go back to.

Just a little foot note. Matilda is also mentioned in "I Could Have Been a Contender". It says "We used to drive all night, all over town. Go Waltzing Matilda, when Matilda came around" I agree with oftehdeepend, I'm pretty sure Brian is talking about "Waltzing Matilda" in this song.