Yhyjyk

So, this takes place in a post apocalyptic world, within the borders of a government called the USC. 30 miles north of The Junction (modern-day Bakersfield) is the JSCF, or Junction State Correctional Facility.

About 2500 prisoners are held within the JSCF. The JSCF is used as a punitive labor prison, where convicts are used to do public works projects, such as building roads and railways.

The USC wants to use the smallest amount of guards they can, to cut costs. So, my question is how could a labor prison be run in a way that minimizes the amount of guards needed to run it?

The current technological level is near that of 1890s America, and the USC has considerable infrastructure

Basic layout of JSCF (please tell me if it isn’t realistic)

Turn prisoners into managers

Many societies across time have used this tactic (Romans for one, Nazis and Stalinists for more modern/malicious examples). Find the prisoners who are willing to enforce rules on other prisoners in exchange for more rights/luxuries/privileges. This creates a type of managerial oversight among the prisoners themselves. It's better to manage than to labor.

Extreme rewards and punishments

Any small violation of rules is enforced brutally. Beatings, tortures, etc. Snitches are rewarded handsomely with accommodations and protection from guards. This keeps prisoners looking for relief an easy way out of their rough situation: betray someone acting out. This type of regulation also keeps prisoners from trying to group together or make a consolidated effort at rebellion.

Group rewards and punishments

If a road or building is constructed well, everyone is rewarded with a day off. If one person makes a run for it, everyone in the group has to work through the night. This reinforces the idea of group obedience.

Keep prisoners occupied

Wake up, work, sleep. Repeat. Give them neither time nor energy to think about anything except their labor. Just enough food to get the work done, and only at the end of the day when they finish the labor. No lunch breaks. Keeping prisoners hungry and desperate for "one more meal" will deplete resolve and eliminate any practical hope of escape.

Although this also breeds intense resentment, there should be a clear path to eliminate or minimize their suffering. Snitching on others, pursuing management over other guards, and performing good labor are simple, safe, straight-forward paths to make their situation better. If there is no (reasonable) hope of escape, but clear paths "up" by turning on other prisoners, then prisoners will pursue those options. Make extreme examples of those who don't.

All these together allow for a few, well-organized, intensely brutal and malicious guards to control a large number of prisoners.

Inmates are the guards. This is the Trusty system.

https://en.wikipedia.org/wiki/Trusty_system_(prison)

The trusty system had designated inmates used by staff to control and
administer physical punishment to other inmates according to a strict,
prison-determined, inmate hierarchy of power...

The bulk of guarding and disciplining of the inmates was performed by
inmate trusties. They also performed most of the administrative work,
supervised by a few employees... Essentially, the trusties ran the
prison system. Highest in the prison inmate hierarchy were the inmates
armed with rifles, called the "trusty shooters." Their job was to act
as prison guards and control other inmates on a day-to-day basis in
the residential camps or out on the field work crews. Next came the
unarmed trusties who performed janitorial, clerical, and other menial
tasks for the prison's staff. Simple tasks, such as distributing
medication, were carried out by other categories of inmates such as
"hallboys.' Inmate trusties enforced discipline within the prison
inmate living quarters (16 different residential camps) and in the
work camps and farms.

Prisons using this system did it for exactly your reasons: limit required paid employees. Trustys were chosen as people who would recognize when they had a good thing going and not screw it up. Oversight was not a priority. American prisons in the South had to quit doing this this way in the 1970s but this system is still used in other countries.