Not necessarily. Uncover a few dozen companies found not complying with the law and prosecute. The consequences are pretty severe. After sentencing, advertise on national television with names of companies and their CEOs in handcuffs.
Yes Strengthen E-Verify. Punish those Companies that break the law and Publish the results and the fines on TV!
Do this a couple of times and You will see results and self Deportation. Get Rid of the Carrot that brings them here to the US.
I am not against Immigration, I am against Illegal Immigration.
That, and work your way "up the food chain". The corporate buyers of those "discount services" are well-aware of how it is they're getting a "good deal".