You don't have to have wide body experience, they took a few A320 guys directly on the A330. But these were exceptions.
Anyhow, Andi, I wouldn't say you don't deserve a 777. But best for your career is to get an upgrading on the 737 and then look further. If you go to Asia as an FO, you will stay an FO, forever. Period. If you are a captain on the 737, your chances for a captain's job in Asia are excellent, and the payment is far better than the one of a 777 FO.
I also know of 737 captains who got a direct captain entry into cargo companies on 747 or the like, with the salary you would expect.
Good luck,
HB