A system and process for separating organic material from solid waste such as material recovery facility (MRF) fines is disclosed. The fines are treated in a press, in which the fines are compressed against a wall having a set of holes. The fines are mixed with a substantial amount of water before being pressed. The water increases the amount of organic matter that is recovered from the fines. A wet fraction, extruded through the holes of the press, may be treated in an anaerobic digester.