This regulator could supply 5V while depleting the batteries all the way down to 0.3V.
Pololu 5V Step-Up Voltage Regulator U1V10F5
EDIT: Note that it is difficult if not impractical to take advantage of some of the low power features available on the SAM3X because of not being able to shut down other hardware on the DUE board. I'm not sure what is the lowest current draw others have been able to achieve.