The primary battery is not completely discharged when the cell phone won't work any more. It has more than enough power to run a small circuit. Let's start with the fact that you still have an energy source in the cell phone, it is just not powering the phone itself.

Maybe when the cell phone dies, it periodically makes a chirp unconditionally until either you plug the cell phone in to the charger, or you physically remove the battery.

the problem here is that there really isnt enough power, the phone doesnt just need enough power to recieve a call and ring, it needs to constantly be awaiting the signal from the cell tower, so it will be using just as much power as if it were running normally, just not receiving outside calls.