Low Input Voltage DC DC Converter to Power LEDs
It really depends on your definition of "extend the time of illumination".

If your regulator pulls enough power to keep the LEDs up at the nominal 3V you may actually reduce the time of illumination compared to letting the LED dim with the voltage. 

However, the best way to implement this type of requirement is to use an LED driver. There are numerous varieties available on the market. 

These work by drawing enough current from the source to drive the LEDs at the current you define to get your desired brightness. They differ from traditional DC-DC convertors in that they are current regulators not voltage regulators.

Again, whether that increases time of illumination is debatable, especially when you now have to add in the efficiency of the regulator.