In Italy, it sometimes happens, like the Agro' brothers who were in the loanshark business in Racalmuto, who asked Salvatore Fragapane (the boss of Santa Elisabetta) to whack Mariano Mancuso in 1992, who didn't pay back enough money and denounced them to the police.
Also, a French dentist Bernard Rouhalde asked the 'ndrangheta boss Santo Asciutto to kill his wife in 1991.
But I never heard about an outsider to hire the mafia in America. Maybe as a favor to labor leaders like Jimmy Hoffa or politicians? But Hoffa was a associate I think, so not really an outsider.