Adjusting private prison contracts may better align their incentives with what the public expects from them as a service.
Prison is intended to either temporarily punish and rehabilitate or permanently remove offenders from society. The more cases that can be shifted to the former, the better.
And to do that, contracts could be based around the post-release actions of the formerly incarcerated. For example, repeat offenses, recidivism, and other lapses would negatively financially impact the prison that should be primarily focused on rehabilitation.
This shifts the focus of the system on improving the post-incarceration outcomes of their former inmates - similar to how ISAs have begun changing incentives in education.