Yes, a third button cell might extend the battery life a little because it would be longer before the voltage dropped to a critical level. But even more of the battery's power will be wasted to internal resistance and in the Arduino's regulator.
A better option would be to add a second pair of button cells in parallel with the first pair. This way, the internal resistance will be effectively halved, leading to less voltage drop.
For battery projects with maximum life, Nano is not a great choice because it has a usb-serial chip which is powered even when no serial communication is taking place, plus it has a power led which wastes power and a voltage regulator which is inefficient. A Pro-mini removes the usb-serial problem but not the other two. It's also a little smaller, allowing more space for batteries. Ideally you want something that will continue to run at a lower voltage as the battery fades, such as a Pro-mini 8MHz 3.3V, and avoid using its regulator by powering it directly with a battery that always provides a voltage within its allowed range.