SD card adapter bought on ebay

Yeah, it supports 5V or 3V3 for power, but the signal lines are not 5V tolerant ... there's nothing on there that corrects that. You need a level converter on all the signal lines. Something like a TXB0104 (or TXB0108). Earlier I suggested this level converter as well, however after more testing I found out it's really slow and hinders fast SPI communications. In order for that to work on an SD card, I had to slow down the SPI data transfer rate considerably. I've since switched to an TXB0108 and can now run SPI at full speed reading/writing to an SD card.