I think the  Experimental Cocktail Club and the  Prescription Cocktail Club are cool, hip places.  I also love  Petit Fer à Cheval in the Marais, which is a tiny place with big time character on one of my favorite streets in Paris,  Rue Vieille du Temple.  Rue Vieille du Temple is dotted with bars, cafes, and boutiques, so if Au Petit Fer is too crowded, there are other good options all around, including  La Belle Hortense, which is literally right across the street (30 and 31 Rue Vieille du Temple, respectively), and  Les Philosophes which is a cafe, but a nice place to have a glass of wine or pernod and sit on the corner to people watch.  

Drinking in Paris is pretty expensive, but here are some in cool and cheap places http://www.timeout.com/paris/feature/selection/paris-best-cheap-bars

Also you could go around Sci 6 Place de la Bastille, there is a small street with a lot of bars and restaurants  Rue de Lappe

Bus Palladium is in 9e, I think, and check out  Le Dandy 26 Rue Pierre-Fontaine 75009 Paris

Go to the bar Name "Dirty Dick"...best cocktails experience ;-) and very close to your place.

Enjoy

There is a whole street of fun bars in the 11th - Rue de Lappe. Its a great spot for people watching and drinks.