Hashi no Shita (橋 の下 means "Under The Bridge" in Japanese) is a place I found in Akasaka-Mitsuke on the main road outside the subway station, a long row filled with hostess bars and restaurants. Unfortunately when I dropped in they were in the transition time between the afternoon "cafe" and the evening "bar", so the owner and staff couldn't spare the time to talk. It's a nice basement cafe/bar, and from the website it seems they serve a lot of food so you can do dinner there as well as drinks.
They also have small live sets of duos and trios, often with no table charge. It's open till 4am so it's a good spot to remember if you miss your train and need some jazz to get you through the night.
Station: Akasaka Mitsuke - Marunouchi/Ginza Subway Lines
Exit: Tamachi Dori exit
Distance from station: 3 minutes