Right now, e-verify is only required for employers that fall into the government contractors bucket.
Now, handyman and landscapers that work for cash will always be the hardest to control. Make 11 million workers fight for those jobs and you'll find their aren't enough of those jobs.
Incidentally, I have a Mexican landscaper in my employ for the last 8 years. He appears legal as he goes back and forth to Mexico often as he's building a "resort" villa there. When he's gone he always had another Mexican crew do our yard that barely spoke English. I'd lay heavy money they were illegal. In the last 2 months, he now has a Caucasian man working for him. This is the first time in 8 years he hasn't had other Mexicans working for him. Maybe he's protecting himself from the risk of finding/keeping illegal immigrant workers?
And in hiring those individuals to do your work, you ARE part of the problem. You have not taken steps to verify your primary is 'legal' and you are all but positive that his sub-crew IS illegal. That means that YOU are not only part of the problem but ALSO at risk of prosecution...