Aside from what is written already.
Electrical engineer/electrician. Pretty simple. People would still need electricity and a person with the ability to keep a generator running is worth their weight in gold.
They fix the lights, the monitors, the hospital equipment, the floodlights, the electrical fence...etc
They would be as valuable as a doctor.
Carpenter/woodworker. Again one of those professions that we need in order to life a relatively human life. Not so valuable but I recon that the ability to work with the materials given would be highly sought after.
Metalworker/smith. This is important. Tools breakdown all the darn time. From nails to knives to shovels to the rest. The ability to produce and maintain those tools is what gives a settlement it's place in the world. Otherwise they are too dependent on others.
Gunsmith. Man oh man. Guns is important. With basic training people can maintain their weapons. But they break. And with no big factories flooding the market with Aks/ARs then the ability to salvage, maintain, and produce the finer parts of firearms insures that you town is safe. It is optional thought. But an important note is that most places would be unique with one having a surgeon the other an engineer...etc and when you get to the larger places those professions become common.
Handymen. Self explanatory.
Administrators. That would cross with the elders/leaders/rulers...etc of the place.
But you need someone to keep tracks of bullets and blankets and water bottles...etc and those skilled enough are as important as someone with a gun.
Sniper. Everyone would know how to shoot a gun. But not everybody can effectively engage targets at a kilo and a half.
They would be rare and most likely would be hunter/sniper or such. But they would be valuable and would spend a lot of time in their own town as you don't risk them. I'd imagine that your settlement has open flat terrain, probably the fields that feed you, around it and that is a perfect for a sniper.
I think the rest does not need a lot of specialization or requires going outside.
Everybody can shovel garbage, shoot a gun, look at a security monitor, drive a car, stand on duty, stack stuff...etc.
But the above stuff requires more knowledge and vital for surviving or thriving.