A simple circuit that would likely work well for you is to use a P-Channel MOSFET with the drain side connected to the battery and the source side connected to the load (MCU and other circuits). Then connect a 47K resistor from the source to the gate terminal. Also connect the gate to an output pin of the microcontroller. Initially the battery will source power to the MCU through the body diode of the MOSFET providing the nomimal diode drop to reduce the fully charged battery voltage to the lower level needed by the MCU.
The idea here is to turn on the MOSFET by having the MCU output pin pull the gate terminal of the FET to GND when the battery voltage has fallen to a level where the diode drop is no longer needed. The ON state of the MOSFET will effectively short out the body diode and allow the microcontroller to receive nearly the full battery voltage.
There are multiple ways that could be used to determine when to switch the MOSFET on. One easy way would be to utilize a A/D converter channel on the microcontroller to monitor the battery voltage and make the switch of the MOSFET at the appropriate level. Since initially the battery voltage would be higher than what could be supported by the MCU's A/D converter input you would put a voltage divider across the battery to feed the A/D converter at a lower voltage.
An alternate scheme could use one of the analogue comparators available on some MCUs. A couple of resistors and a small diode used for setting a reference level would bias the comparator inputs to be able to detect when the battery voltage has fallen to the level needed for switching the MOSFET. On some microcontrollers with the comparators the output of the comparator is available on an I/O pin. It may be possible to connect this directly to the MOSFET gate without having to use another separate I/O pin to drive the MOSFET.
There may be a cart before the horse scenario here with the ides I describe. It may be necessary to use an external circuit to switch the MOSFET because the MCU may not be able to run properly up till the time it gets around to using software to switch the FET. This would be particularly true with the A/D converter scheme. On the other hand on-board comparator it may just be possible to make it work based in just how the default configuration of the comparator comes up at power on. If that cannot work then the MOSFET switch logic would require some external components to be able to do its job.