Best choice to power up a ESP8266 with a battery

Dear all,
I'm aware a long time has passed, but I would like to update this topic with the final approach I chose.

The board is currently being powered with 4 AA batteries (rechargable, even if I have read that not-rechargable would be a better choice).
The dc converter is a "POLOLU-2842 Pololu 3.3V, 500mA Step-Down Voltage Regulator D24V5F3" that has a very low quiescent current (something like nano Ampere).

The board (deep) sleeps for 5 seconds than check the inputs for HIGH levels. The HIGH ref is provided by a pin used as OUTPUT, this way I can use current only for the time required to read the pins.

Last but not least, I have removed/disconnected the onboard dcc converter and usb serial, so that the board, when deepsleep, uses a very low amount of current.

This setup could ensure a good run time on battery. I have made some esteem but I think the best analysis will be done in retrospect: the board and the circuit is up and running now, I will just observe how much time it will take for the batteries to go flat.

Thank you all for you help!
MJ