The voltage drop across the diode was .3 volts, so the 01s module was getting 3.65 volts. I figure as the battery wore down over time the voltage would drop and get closer to 3.3 volts, but as it was the 01s module ran fine at 3.65 volts.
I then put an amp meter in the circuit and got a reading of 25 to 70 milli-amps over the space of 5 seconds. These 8266 modules obviously don't exhibit steady current consumption. I'm sure the blinking LED (to indicate wifi connection) played some role in the fluctuation. I've flashed the module with tasmota sonoff-sensors.bin. I imagine that different firmware will result in different operating current draw.
But the initial power-up current draw (which my meter is not capable of measuring) must be huge to blow a 1-amp diode. Any comments?