DISCLAIMER: This is just a suggestion. No hate for this anwser please.
You could use a capacitor. A super capacitor (high capacity) with a proper joule theft circuit for your project could power the arduino for propably another 5min even if main power is cut off, plus you do not need any extra power source once the main source is cut off. It can be useful if you need to change the battery or move it to an other location but you do not want the module to be turned off.
const int led = 13; //led pin
const int sensor = A0; //connect this your power supply, use a voltage divider if necessary
//this will act like a volt meter
int V = analogRead(sensor);
V = map(V,0,1023,0,5); // the 5 is replaced with the ammount of voltage that you supply the arduino
if(V < 1) // when the power is below a point or simply disconnected, the capacitor
//will kick in by itself because the Arduino will draw its power
while(V < 1)
V = analogRead(sensor);
V = map(V,0,1023,0,5);