The US is completely different to the UK in prisons and in sentencing, the structure of the organisations is so different that it is very hard to make a comparison.
The only thing that can be said is that both places have very hierarchical organisations, and probably have similar accelerated promotion schemes.
There will also be certain specialisations that are extremely popular, such as dog handlers, and PE instructors, specialised search teams, in the US there will be firearms trained staff, in addition there are staff trainers such as for control and restraint.
There will also be many associated roles that will not be part of the correctional hierarchy, from procurement, safety specialists, counsellors, maintenance.
In other words there are a very wide number of other roles, and it depends upon how the management line is structured as to how it is possible to move onwards or sideways to other jobs.
It really depends upon the sort of person you are, handling prisoners is likely to be the longest and most numerous chain of command - so that’s where the career prospects are likely brightest - that’s where the prison governors will mostly drawn from. However there are different circumstances in handling prisoners, some are much more adversarial all are controlling but in different ways, from physical through to my type of work which is about getting prisoners to carry out your instructions in a (slightly) more cooperative way.
Being a prison governor, at any level - there will usually be several different levels of governor - is an extremely responsible role, lives depend upon the decisions that you make , other staff can submit reports and recommendations, but its the governors who are the responsible persons. When it goes wrong they are the ones held accountable at inquest, they are the ones who make public protection final decisions about what to do with offenders.
In addition the offences of some prisoners can amount to being of public confidence issues, what happens to them can become small p political, and a full on riot is always likely to make it into the news, and questions may well be raised at national political levels - I personally know one prisoner who claimed compensation and won, it directly resulted in questions being asked in parliament - I cannot go into the case, however it was extremely newsworthy.
Frankly that’s not a level of responsibility I would have.
So in short a career in prisons can go from simply night checking patrols through to national executive levels, and yes it can happen for a basic grade officer to make it all the way up to Chief Executive, I have seen it done.
You can pretty much go as far as you like, the only thing you will be extremely unlikely to do is to use it as a springboard into political office in the UK, however the US might be different.