I am not understand whether SD card reader is a device connected over SPI (as so you are using certain communication protocol to communicate with this SD card reader that is managing flash file system) or you are using SD card itself in SPI mode.
However, we still don't have a solution for either option.
In the first case you could create communication protocol according SD card reader manual.
In the second case you could possibly use some public source codes (possibly operating system like FreeRTOS) for flash file system use. Unfortunately we don't have any example or library for that.
Also I would apparently consider to use some device having SDHC controller as in SPI modem bandwidth is limited.