Pixel Battery Saver (free) turns off some of the pixels in the AMOLED display, reducing power usage. The screen is a bit darker, but even turning the brightness up a bit to compensate it seems to use less battery on my S4. It's worth a shot, you can toggle it on and off easily, or set it to come on once your battery drops below a preset level.

Note: hit "cancel" to the first prompt when you run the app. That gets rid of advertising I think.