This first thing to do when you want to flatten the sole of a plane is to check whether it needs flattening at all. There are two reasons that you might choose not to flatten the sole 1. It is too far gone to be worth bothering with 2. It is perfectly flat enough for your purposes.
I employ two tests to before deciding what to do. First, I make sure the iron is fully retracted and I lay the plane sole down on a flat surface (in this case a ceramic tile that I’ve checked with a straight edge). I then use a 5 thou (.127 mm) feeler gauge and se if I can slide it underneath the plane at any point. If I can, then I probably won’t go any further with flattening, I’d consider turning it into a scrub plane, for really rough work (something that I might do with one of my #4s or my#5½).
As you can see, the gauge only goes under a couple of millimetres. This is because the edges of the sole have been feathered to minimise marking of work pieces. That is perfectly fine and to be expected.
The next step is to see where on the sole needs the most work. Simply draw a series of lines with a marker pen across the sole, especially around the throat (or mouth).
Using the ceramic tile from earlier. I stick down two strips of sandpaper, one of 80 grit and another of 120 grit. Then it is just a case of “planing” the sandpaper, starting with the 80 grit, making sure that the iron is retracted.
Check to see what has happened to the pattern of stripes. They will wear away in the places that are in contact with the sandpaper. The areas of concern are the toe and heel and the sides, and around the throat. Id they are all worn away, then it doesn’t really matter if there is a hollow in the middle. Ideally, you want the ares immediately in front and behind the mouth to be clear of the pen marks, but for a jack plane I’m not too worried.
As you can see, we’re in pretty good shape, so it is over to the 120 grit to polish the sole up further. You may want to go through the grits to polish up the sole even more but, again, for a jack plane I’m not concerned with that.
The other thing you might want to do is to feather the edges. This can be done by placing a straight edge underneath the long edge of the sandpaper and taking some more “planing” strokes, flipping the plane around and doing the same for the other side. This will mean that the long edges of the sole will not mark your work, although , yet again, this is not critical for a jack plane.
With the sole sufficiently flat, its time to disassemble the plane and clean up and de-rust the smaller parts.