Check the voltage from the hot side of one outlet to the hot side of the other. If you get 220v, your generator is split-wound, and yes... you can pull 220 from it. If you still get 110v, the generator is a single-coil type (both outlets are powered from the same coil), and no you can't.
Best bet is to just get a bigger gen head.
Regarding that 220 adapter, you can make one easy enough, just remember that your maximum allowable amperage is 1/2 at 220 than at 110.