Edit: Modified a second module, also running fine now
In some circuits, like the Olimex one, they've pulled this pin to 3.3v with a 12k resistor. This is pretty common practice for mcu reset inputs.
So for the ESP-01 I would suggest adding a resistor from the RST pin of your socket to 3.3v, or even just joining RST to 3.3v if you're not going to need a reset button, instead of having to directly mod your ESP-01.