Was it the only thing Bill was able to salvage and take with him when his wife evicted him?
Or was it the weapon Mrs. Bailey used to actually chase him out?
“You remember that rainy evenin’
I threw you out… with nothin’ but a fine tooth comb”.
The author clearly needed an editor!
The role of it is, it rhymes with “home.”
Only upon seeing this thread again, an hour later, I realized you’d given me quite the earworm. Thanks a lot. I may check back in with you tomorrow. Meanwhile, I wish you, I don’t know, the theme from Gilligan’s Island. 
Back in my student days, some guys took to keeping metal rat tail combs in their back pockets. The point was kept sharpened. School officials eventually got around to banning them.
Maybe “fine tooth comb” was originally “rat tail comb”?
(They are available on Amazon for $5. That’s for all steel. I don’t think the plastic ones with a thin metal rod handle would be very effective at kicking Bill out.)
I always thought the fine tooth comb bit was a euphemism for " when I kicked you out with nothing but the clothes on your back". Now I’m giggling, picturing Pearl Bailey shooing Bill out the front door with an actual comb. I like it.
It puts a different possible spin on the song. Maybe the singer is glad to be rid of Bill but wants him to stop by one time so she can get the comb back.
He gave her crabs?
LOL.
I finally heard Baby Shark the other day. It stuck in my head for several hours - mainly because I’d heard it was such a terrible earworm. I seem to have forgotten it by now.
It’s a small (musical) world, after all.
*pierces eardrums with sharpened rat tail comb*
Sorry - the earworm is actually in the brain. That won’t help 