The next stone in the standard polish would be Binsui-do, a natural stone, but the one I have is much finer than the Kongo-do, I'd call it about 600-grit equivalent in sandpaper. In my honestly fairly extensive grinding experience, while you can go from the coarse -- I'm revising my estimate of the Kongo-do stone's roughness to ~120-grit -- to that fine, you're going to be spending a long, long time working out the heavier scratches. If I was stuck with just the stones you can get as a kit, I could feather out those coarse scratches by varying the scratch angle, using the nagura to keep plenty of mud on the stone, using a very light pressure, and hoping I didn't end up with stray grit causing even more deep scratches.
Fortunately, I have a number of other stones, manmade and natural, that fit between these stones. I'm starting with a #800 artificial stone, which is removing the scratches nicely without leaving too heavy a scratch pattern behind, followed by either the #1000, the #1200, or the queer creek stone before moving on to the Binsui-do, which is about a #2000. After that, I think I'll try to stick to natural stones. I have a block of blue Aoto mountain stone, about #3000-4000, and then we'll see. The idea is not to get a mirror finish, that would be easy to get just with a buffer, but to bring out all the elements of the hamon and related effects. "Opening a window into the steel" is how it's best been described to me.
With the 800 grit stone, I'm working in a circumferential pattern, which allows me to clearly see when the coarser scratches are gone, as well as allowing me to blend in that last bit of clam edge while bringing down the bevels to a wire edge, the final stage of the base polish. This artificial stone is a bit harder than the last, which is fine for this application, but it is leaving a very "bright" finish, while a softer stone would leave a more matte, but still smooth, finish. I will use this stone until I have removed all radial scratches, and have a wire edge all around.
It is fairly hard to keep all the angles set while applying downward force while also moving the blade back and forth, and I am feeling pain in my finger joints from the pressure. Fortunately with the heavy work done I can now use less pressure, and that should let me have longer sessions before having to take an arthritis break.
Also, the chair I've been using for this polishing had all the caning mysteriously shredded, though both cats disclaim any responsibility. Fortunately, years ago I read an article on caning chairs, and I've wanted to try it out, so I'm about halfway done after half an hour to re-caning it, using the heavy cotton cord used in theater to tie up/down cables and lights. Since there's a cushion to go on top, I'm just building a support gridwork instead of a sold weave, but so far so good. We'll see if it can hold up to the cats.