FPGA Configuration Flash as General Purpose Memory After Boot
If you are using the FPGA with an industry standard type of SPI Flash chip then it is possible to use the SPI Flash chip in the way that you describe. The way to do this with the most user control is to put a small MUX Chip between the FPGA and the SPI Flash chip in the path of the SPI Flash control signals. The other side of the MUX is then connected to alternate user activated pins to support an alternate SPI interface to the SPI Flash chip. You would then program your own interface into the SPI device to access as desired. 

On the other hand Xilinx has a detailed App Note that shows how this can be done in a more generic way without the added MUX. See http://www.xilinx.com/support/documentation/application_notes/xapp694.pdf.

A more directed document for the Spartan 6 FPGA family is here: https://www.xilinx.com/support/documentation/user_guides/ug380.pdf