I just tried using the palette and immediately I began to get the feeling as if you didn't use the palette at all so far and I'm definately not impressed by the way it's setup. Alot of colours have similar luminosity and some very common colour ramps simply can't be achieved.

While making this tree I found important colours missing, things not combining.

At 1: the gray doesn't mix with the green and yellow as it is equally luminescant, why isn't the yellow brighter than the green. Hue blending from gray/blue or even purple through green resulting in yellow is much more common than a ramp that uses a bright green at its end or a ramp that uses yellow as its center colour. The colours at 2 are all the same when put next to eachother. They can't be combined as neither stands out across the other.
Then at 3 you have a dark brown that is between 2 blues, I don't want to use that brown unless it's lighter. Taking the brown at the top right of the colours I arranged around 3 and desaturating that, then desaturating the blue directly below slightly would make up soemthing much more useful. Then the grey needs to be fit into that with a different luminousity.
Now I adjusted the colours to fit my tree but most of all keep making stuff with the palatte, make it pretty by adjusting the colours and then see if you can change the colours on all images simultanously while retaining their beauty. It may be that some images will go out to use different indexes of the palette but as long as with some colour swapping they'll retain their readability and prettyness your palette is improving. Draw alot, make use of it alot. That's what makes it a 'generic' palette.