You have just defined a path to amnesty.
The problem with the present set of laws is illegal entrants do not have visas ~ yet, the courts have given them various rights to appeal their deportation orders.
Give them a visa. It categorizes and classifies them and lines them up for treatments already well exercised under the law.
In effect, they ask for a very limited visa when they step over the line. Violating the terms of that visa gets you put in prison.