In Paris For A Month? Out of the Way Things to Do?
what kind of stuff are you into? On the art tip there's the galleries on the rue de Seine and smaller museums you might not have tried (Musee Maillol, Fondation Cartier...). For going out depends if you prefer bars (anything 19th-20th is good) or clubbing (OK this one kinda sucks nowadays but you could always try le Social Club or le Glazart. Also 2Many DJs will be playing their yearly Xmas gig on the 15th in the Villette hall, which is generally a lot of fun). Food I'm not even gonna pretend I know what I'm talking about.
List your main interests and I'm sure ppl will have more ideas. I left a few years ago but I still go back 1-2/year so I have a rough idea of the new stuff
in that case you might enjoy the oldest tree in Paris haha (sq Rene Viviani, right opposite Notre Dame on the Left Bank). Sounds like you'd enjoy just picking a neighborhood and walking around, which is always a fun thing to do in Paris, even the touristiest of areas will have interesting spots that feel off the beaten path. Not to harp on about that dang tree but close-by you also have St Severin which is a nice haven of tranquility during the day. If you like movies that area has the best art-house theaters in town. Le Reflet Medicis on rue Champollion is a good one (I used to hang out at the bar opposite all the time, it's a good option if you're in the vicinity and want to avoid tourist prices) but there are tons more.
Les Abbesses is another area you might want to check out. It's quite hilly and ridiculously quaint. It's close to Pigalle so a good pick for going out later on.
Will post more stuff this weekend if you're interested
this isn't exactly the best time of year for this suggestion, but if you're looking for a nice park to chill in you should try the Parc des Buttes Chaumont. It's got a very cool 19th c. Romantic Anglo-Chinese garden vibe to it that's unlike any of the other parks in Paris, and has a few hilly lawns that are great for sitting down with friends or making out with a new crush.
On the NW side of the park there's the recently renovated Rosa Bonheur which is a nice bar/restaurant. I've never tried it but feedback from friends was that prices are stupid high and the crowd is filled with mostly awful neo-hipsters. So in a way it's a perfect incarnation of the new 19e arrondissement, which means a visit could be interesting from a sociological POV.
Close to it there's one of my fave hidden secrets of the city: the rue Georges Lardennois. A great view of Montmartre and a random vineyard in the middle of Paris, what more can you ask for...