To clarify, I was giving an example of both volunteering and hiring. Hiring someone doesn't change the economics of it, maybe even making it worse because you now have the state spending someone on a law breaker. Does it spend as much money to hire help for those who don't break the law who could benefit from the services offered?
The real main difference is when people stop viewing prison as being for punishment, but that goes back to the original question of why is rehabilitation incompatible with punishment. If prison is seen as being for rehabilitation and containment for safety while rehabilitating then the comparison is different (though ideally the services offered shouldn't be better than those offered to people who follow the law).
I believe the biggest differentiators are inherited from the fact that many countries who aim for rehabilitation first are also countries with free access to the kind of help many people incarcerated might need: free access to a psychiatrist or doctor, anger management therapy, educational help, etc. So not only are most of the money already earmarked for this but the extra it does costs to offer the help to inmates that aren't allowed to leave the prison to access these services are likely less than the cost of a punishment-first system. The lower recidivism rate added with the fact that these systems also rarely label people as former inmates, which often would block access to well-paid jobs that pay more taxes, must be a lot cheaper but... I can't be arsed to look it up.
As an example from around here (Denmark) if someone want to hire a person they do not by default know that this is a former inmate and they will only have access to this information if the job somehow is related to the crime committed in the past. You could ask the job applicant to bring a form from the police but if the sentence is unrelated to the job (or a set amount of years have gone past if it is a lesser crime) the form will be completely blank. Only related crimes will be shown.
All in all I believe that such a system is also punishment (few like to be locked up) but this rehalibation-first-system will punish everyone else less in the long run.
The real main difference is when people stop viewing prison as being for punishment, but that goes back to the original question of why is rehabilitation incompatible with punishment. If prison is seen as being for rehabilitation and containment for safety while rehabilitating then the comparison is different (though ideally the services offered shouldn't be better than those offered to people who follow the law).