There is a demand for dual-rated instructors, but for now it's easier for the flight schools to give the required training to FAA instructors, rather than dealing with the visa process.
The 1 year visa, may not be valid if it only allows you to work/study in the field you have studied. i.e. if you have a degree in BA you may not be allowed to work as flight instructor.
IMHO the market in the US is hard, but since there's such a huge aviation industry, there is always something going on. In Europe the competition is as fierce, but aviation is a smaller industry, so it's always more difficult to find an opening.