ICE will be very unwilling to release someone with a "final order" but the non-citizen can petition to the Detention & Removal Office (DRO) to appeal to their humanitarian side to request release. If you are one of the few to be rewarded with release, it may be on a very high bond, ankle bracelet and/or close monitoring.
Will you be prosecuted criminally?
One option that is unfortunately considered and more so in recent times is if the foreign national. S/he could be prosecuted federally for criminally reentering the United States after being previously removed (and if found guilty serve additional jail time). It varies on jurisdiction if ICE exercises this option through the U.S. Attorney's office, and often times any prior criminal history will factor into the decision making process. This is the worst situation as it will oftentimes mean significant additional jail time in a Federal facility.
Reinstatement of Removal
Whether ICE prosecutes you criminally or not, they will seek to reinstate your prior removal order, meaning execute it once again and send you back. You will not be entitled to another hearing before an Immigration Judge. To prevent a subsequent removal, there are several options available: 1. Apply for Haitian Refugee Immigrant Fairness Act of 1998 (HRIFA), or section 202 of Pubic Law 105100, the Nicaraguan Adjustment and Central American Relief Act (NACARA); 2. File a petition for review in the Circuit Court challenging the Reinstatement; 3. Apply for asylum type relief known as withholding of removal or deferral under CAT Convention Against Torture. If you cannot succeed on any of these, the noncitizen will be deported again.
Additional resources provided by the author
Overall, this is one of the most difficult and complex types of immigration cases that I have handled (and successfully).