I seem to remember in Proulx story that Ennis chose for himself a horse named "Cigar Butt".
Suddenly it dawns on me that this bit of trivia conceals a deeper meaning. I become aware of it when I consider Jack's comment to the bartender, that he is not rich enough to have a horse.
I see a connectiong between "cigar butt" and "stemming the rose" and Jack's comments to the effect that there is no horse that can throw (or quit) him.