Until you jump up to something really fine like chromium oxide/other honing compound or a sub 1 micron stone like the 13k, its hard to tell much improvement from other stones above the 6k after a palm stropping. The edge off my 6k is improved by my Naniwas Snowwhite 8k and I used to have a sigma 10k and that improved the edge also. However, the improvement is not easy to see in standard sharpness tests. By that I mean you will be able to pare softwood endgrain off the 6k and when you go to an 8 or 10k you'll still be able to do it with about the same ease. However the edge is sharper. You see it in the form of edge longevity and better result on less well behaved woods.

BUT Only when you go to something significantly finer than the 6k do you really notice the improvement. If you go to the 13k you go from cleanly paring softwood and leaving nice bright finish to sliding through endgrain that glows after the cut is made. Honestly unless you are going to go to something really fine like a sig 13 or a shapton 15 or a very fine honing compound I wouldn't bother going with another stone above the 6k. Until you get to something really fine the difference just isn't that significant, and if you know to let the 6k dry to work it finer and then palm strop it pretty much will make it better than or equal to most 8 to 10k stones in terms of how it performs on wood.