Power is Volts times amps. There is nothing you can do to change that. If you drop to 3.7 V, you will increase the amps relative to that. The battery won't last any (appreciable) amount longer or less.
To increase the time the device will run, you need to increase power. Since volts are limited, your only option is to increase amps. You can run any combination of volts/amps you want, but you will require more batteries. Of course, the other option is to reduce power consumption. Arduinos have several options to do that using sleep methods and sometimes combined with hardware modifications and other voltage regulators. You don't go into what 12V hardware you are using but it may also have some method to save power.