Power an arduino nano with the best efficiency

Hello everyone, I am learning arduino and I wish to make a kind of probe that measures temperature and humidity (with a DHT22) at regular intervals. I want it low-power so I will use a TPL5110 from adafruit to power the arduino every 30mn. I want to use an arduino nano for that project (because I have one). The problem is that the TPL can't be supplied with a voltage up to 5V , so it means the nano can only be supplied at 5v , wich is not a problem if I use the usb port. But I wonder how can I have a 5v power source. 5v batteries don't exist, and I have seen that a voltage regulator is not efficient and draw current even when the project is disconnected (in addtion of the current draw by the TPL5150) , wich is not acceptable for my purpose. Do you have any ideas on how can I get a 5v power supply with the best efficiency ? Thank you !

4 x AA alkaline ---- -->|---- -->|---------------Nano 5V pin 6.4V new 2 x 1N4003 diode 5.0V

4 x AA NIMH rechargeable ------>|----------Nano 5V pin 5.6V, full charged 1 x 1N4003 4.9V

Thank you for you answer. I already own several 1N4007, I'll try with them