Good post from Koru with some good ideas, but as an American citizen I still advise to stay away from U.S. contract agencies. Therefore GAP for KAL and HACS for Japan are probably not the best idea. Not that there is anything wrong with those agencies, but life is far simplier if you do not get a 1099 form from your employer. I would reccommend either Cambridge Communications ( very good reputation ) for KAL or PARC ( also good ) for Japan if you are a U.S. citizen.
I would avoid Rishworth based on their reputation.
All that said, as a 737 First Officer your options are limited. It's far easier to get a job as a captain on the foreign contracts. Others that hire first officers are Cathay Cargo, Viva Macau, Gulf Air, Air Arabia, China Airlines, EVA Airways, Jazeera Airways, Saudi Arabian, Saudi Aramco, as well as those mentioned by Koru. You might try to get a recurrent session in the CRJ and that could put you in a position to get a CRJ captain position somewhere like Arik in Nigeria as well as some of the Chinese operators.
Typhoonpilot