Issue “Temporary Work Search Permits” (TWSPs) to foreign would-be workers at entry, without tentative jobs. By TWSP´s expiration, the bearer would: (1) find a job, (2) leave the USA if no job is found or (3) request an extension of time.
These workers would be free to travel abroad and they would be a new legal group of people, in addition to: (a) U.S. citizens, (b) green-card holders, (c) people with tourist or business visas, (d) people who do not need passport visas, and (e) people with other statuses -like international students and diplomats.
TWSP holders will not exceed the quantity the U.S. economy requires (by labor demand and supply): They will not want to be unemployed in the USA. Besides, since only criminals would try to cross illegally the borders, TWSPs facilitate security.