A light-weight puffer coat is more than enough to get you through the chillier mornings in Madrid, but not so warm that you have to take it off in the afternoons.
I hadn't packed mine, and I wouldn't say you'd strictly need it to stay warm if you're used to a colder climate, but they sure are cute.
Clean lines and basics colors, and you'll fit right in! If it's warm enough, I like a nice, classic, dark blazer to finish off the outfit and stay warm.
A light trench coat or classic Barbour jacket would be perfect.
If you want to wear a dress or skirt (not my preference for tourism, but to each her own), be sure to include some warm tights.
I'd also include a pair of 'formal,' non-denim trousers for nighttime wear.