Briefly before going to bed I stopped to ponder on an alternative look, or possibly alternative class all together, seeing as the Operative really wasn't fitting it. I then began to think what else would someone who's not wanting to be noticed would wear, high collared coat, hat, possibly sun glasses, and I began to ponder we're I've seen that look before that would be seen amongst shady characters. And then it hit me.
Maybe not strictly the Mafia, but a similar sort of looking shady character carrying a violin case with a Tommy gun inside it, or possibly even another deadly weapon concealed in his large coat. Furthermore this character fits in with the rest of the Team Fortress 2 family miles better than the Operative did with his otherworldly weapons, gadgets and tactics. Additionally he doesn't necessarily need to be in the Mafia, he could be in some other organised crime agency, or even an undercover cop from the 50s.