Murray Humphreys, who wasn't officially inducted in the Mafia, was also basically a consigliere for the Outfit who had tremendous influence.
...and so did Meyer Lanksy, who was, in effect, Charlie Luciano's
consigliere, and who was a close adviser to other Dons, including Frank Costello.
Re. Hagen: One of the nice things about being a Don (especially the most powerful Don) is that you can do just about anything you want to--including naming a college graduate/lawyer/German-Irish/non-muscle person as your
consigliere.