Hello eexpress,
I think that your problem is that the 93C56 device is not actually a SPI compatible. The SPI uses multiples of 8-bits. However, with the 93C56, assuming you are using 8-bit organisation, a read command consists of one start bit, two command bits, and 9 address bits, a total of 12 bits. Not sure how you organised this with the SPI. Eight data bits are then returned, but these are preceded by a dummy bit - if this is what you are observing, you are probably one clock pulse short. So the situation becomes very "messy".
I have previously used the 93C56, but have always used the bit-bang approach, without any problem. Of course, this is considerably slower than the SPI. I could provide assembly code if you are interested.
Another solution is to use SPI compatible EEPROM, such as the 25Cxxx series. The smallest is probably the 25C080, an 8-kbit device (four times the capacity of the 93C56). This is probably the cleaner method.
Regards,
Mac