Maybe there is a workable solution with some history: We have had two additional paths for foreigners in the past, one for temporary workers and one for citizenship, and they both worked quite well.

I'm talking about California's Bracero Program for Mexican farm workers, and foreign recruitment for the Armed Forces. Maybe we could combine ideas from those two programs to create a Farm Corps (FC) that would give existing illegal immigrants and new applicants work assignments under humane conditions. The FC would contract with farmers, packing plants, etc. to offer them minimally trained workers at a slightly sub-minimum wage rate. The FC would also supply uniforms, barracks, mess halls, and sanitary facilities for the workers or require similar standard facilities for workers who live on or near the work sites. (I would really prefer my salad ingredients being picked by somebody who washed their hands after relieving themselves!)

FC workers would be free from fear of deportation (as long as they behaved reasonably well), could safely send money home to their families, would have humane working and living conditions, and after a number of years (3, 5, 20?) with a clean record could enter a citizenship program. That program could include English and US history and culture classes, so the resulting free citizens would be assimilated as well as any long-term resident immigrant can be. If we can make such a program much more attractive for the workers than the current situation, then we could get very high participation. That would also solve many of the security problems with the current mess because the FC would know exactly who they had, and where they were working and living.