Sorry to sound a bit harsh
No foreigner can get jobs in the US, especially when it comes to jobs in aviation and especially Pilot positions. The US has a tough policy on foreign workers and I think it is fair for the EU to protect its labor market especially in this fragile and cyclical industry.
Getting a JAA license won't help if u're not legal to work in the EU and to be honnest with you, there are enough pilots looking for jobs and too many won't make it to their dream job...look at the wannabee section and you'll see how difficult it is. My advice to you is stick with your FAA license, build time and even if it takes longer, you still have many possibilities to work as a Pilot!