First off, potato soup isn't potato soup without ham. You can do this with a slice of ham, but you'd have to do it differently.
I start with a half ham. When I pick a ham, I only buy one that is very pink. If it's faded, I don't care what they tell you about "Oh, it's just from being under the light" - it's not going to be as tender and flavorful. Also, the center of the exposed bone should be soft. They now put a piece of plastic over that so that you can't tell. Gee, I wonder why they do that.
Boil the ham for about a half hour, then discard the water from it, as it will be too salty to do anything with. Then continue boiling the ham. I will boil it for about two hours, until it's falling off the bone. This is because I'm not interested in putting it in the oven and making it all pretty. Do as you like with the ham, just save enough for the soup - about one thick slice.
During the last hour of boiling, add one or two cut up potatoes to absorb salt. Discard or eat these, but don't use them for the soup. You can also add cabbage and/or celery to eat or just to flavor the stock.
I, at this point, boil just the bone alone (in water), to get extra nutrition and flavor out of it. I also eat the marrow. Then I add this stock to the other.
After removing everything from the stock, boil the carrots and potatoes. I use fingerling carrots and cut the potatoes to about half-inch cubes. You can also add cut celery. Once the potatoes are tender, add flavorings and ham chunks.
For me, flavoring means commercial cream of potato soup. I have no idea what my mother used. Also, add enough milk/evaporated milk/cream to make the soup white (about 1 or 2 cups). For one to two gallons of stock, I use two cans of condensed soup.
That's all folks!
Just one day's work on this one.