To be honest, I'm just guessing at the colors here, but this is the thought process I went through. I post this in the hope that one of the better pixel artists can provide a better method.
1. This is the first sprite. Red, black, white. Maximum contrast.
2. One could simply add in-between colors. This would give better results for the buffer colors, but that only really works well when there is enough pixels. I don't think it would would work for the very small arms and legs of the sprite.
3. Adding another hue is a good choice to keep the contrast, but it makes 10 colors.
4. Using the mid-brown that you had as the new hue. This brown is pretty closer to pure red than black or white, so eliminate the red/brown combination.
5. Reorganize the layout and see that there is basically 3 ramps. A red, a brown, and a gray. But these are all single hue ramps, and single hue ramps don't look that great.
6(a). Shifting the white to a warmer yellow, and the black to a dark violet adds hue contrast to the gray ramp. These were random choices. Something warm vs cold(ish), while staying light vs dark, and staying in the reddish hues so that they would ramp nicely with the reds and browns.
6(b)Halfway between violet and yellow would be an orange-red, but this starts to look to close to the red and brown colors already present. I flipped the former gray around to the other side of the color wheel to make it a blue gray. It's still midway between the yellow and violet, but pushes into the cool side of the spectrum so it has stronger contrast with the reds and browns.
6(c). The dark brown and dark red are close together so I removed one to reduce the color count down to 8 colors.
7(a). Aligned the ramps by lightness, from dark to light. If these colors were grayscale, now they would be separated by an equal distance (roughly where they sit left-to-right on this image). But I'm not sure if this is the best pattern. They could also be put in a V-shape, where the reds are lighter and darker while the browns are mid tones (or vice-versa). This pattern only adds one ramp, dark red - brown - gray while a V-shape would add two.
7(b). To increase the contrast between the red and brown ramps, I decreased the saturation on the brown ramp colors. I think desaturated colors work better for skin tones anyway.
Final result is 7 colors with 4 ramps. Sprite has 4 ramps (face, shoulder/boots, torso, pants), so this works out ok. I know there are better methods out there, so please don't give this too much weight.
Tourist