Yes, pin 10 is set to output.
However, I think I've just solved the problem. The SD card module was not releasing the MISO line when the CS was set to high. I thought that it might be related to the fact that SD card runs on 5V and ethernet module on 3.3V, so I just added 220 Ohm resistor between SD card MISO and Arduino and apparently it works now both for ethernet and SD card, though I'm not sure if my reasoning was correct.
Anyway, thank you for your help.