I did try connecting it from the 3.3v pin on the module to the 3.3v on the UNO. It didn't seem to make a difference.
Correct it won’t. The voltage into the SD card reader is irrelevant as there is a 3V3 regulator on the card. What doing this does not do, is to change the voltages of the signals being produced by the Uno it is still too high for the SD card.