The easy way for me to rack up contribution is to start my own Hard mode fight. Try to rack up at least 1 million contribution in the first few turns, as the gain rate sinks like a rock afterwards. Then send out reinforcement requests.
Alternatively, I've occasionally been able to get decent contribution from joining other fights, but I still haven't figured out the system. Couple more hours of Hard contribution farming tomorrow should get me to 500 coins.