Voltage stabilizing circuit
You need some capacitance, as energy storage, a harvester to charge it, and a converter to use it. And some low power control circuitry to manage the whole thing.

Your ESP32 uses about 1 watt when running. If it runs for 10 seconds, that's 10 Joules. Let's choose to use a buck converter to get between the energy store and 3.3v, so your minimum capacitor is allowed to be 4v or so, let's say 5v to give a bit of margin. Allow the capacitor to supply up to 10v, a 2:1 voltage swing uses 75% of the energy stored, a reasonable figure. So you want a capacitor that will store 13.3J at 10v. A buck converter isn't 100% efficient, so we inflate this by another 10/15% to allow for losses, to get 15J.

\$E=0.5CV^2\$ so C = 300mF (300,000uF), which is big, but not too big for a supercap. You could use a smaller value if you chose to charge to a higher voltage.

Now you need a harvester circuit, to use your 1v to 16v inputs to charge this capacitor. This is where you need a lot more information about your power source and its power capabilities. Texas Instruments do a lot of power harvesting converters, so read their application notes.

You may need some low power (read CMOS) amplifiers or references to monitor the store, to trigger the ESP32 wake-up, or to see if it's OK for it to wake up when it asks whether it can.

Notes on the energy store. You will find some fairly cheap supercaps designed for memory backup, which may not be able to deliver the 1 watt needed for your ESP32, check the power output of any you buy. You may be better off with a LiPo. Choose one large enough that swinging 15J barely changes its state of charge. This would give it very long life, NASA have found that charging their cells to a max of 3.92v allows 8year+ life in satellites in a 90 minute light/dark orbit.