I have long advocated that we put bounties on illegal aliens. Specifically those with court orders or who have overstayed their visas. Put them up on a public website, Wanted $500 Bounty. Include whatever information is available. Let the American people find them, root them out, and then ICE can deport them all. It'd be good win-win for everyone.
Last night I eavesdropped on a border patrol agent discussing the issue. He said policy for visa overstayers and illegals who have been in country for a long time was to send them to a hearing. The judge then typically grants a permanent stay on deportation and orders resident status. The authority to decide whether to deport or grant a hearing is in the Border Patrol. The authority to define “long term” is also in the BP, and his supervisor let him know 2 weeks is now considered “long term”...