Even if it is a group fare that EK can't officially touch, they can get you on an earlier flight if you speak to someone who knows the system.
Through a strange course of events, I ended up at the airport 18 hours before my flight. The girl at the EK desk said she could do nothing as it was through a travel agent. The girl at another desk was great and got me on the next flight - I think she checked me in (18hours ahead) then did a Transfer to the current flight.