French Restaurants on Madison/60's
I'm not sure there is a French restaurant in the 60's right *on* Madison.
Daniel is on 65th, b/t Madison & Park. Though it's obviously a destination restaurant, someone living nearby could call it "local."
For something much more casual in the general vicinity, Sel et Poivre, on Lex, b/t 64th & 65th, is a charming French bistro with very tasty food.