I think it's smart to use a color palette across a scene, but your color palette should definitely change from scene to scene. The colors you use in a brightly-lit outdoor scene will be totally different than the colors you use for a dark nighttime scene which will be totally different than the colors you use for a murky underground scene, even if the characters are all wearing the same outfit. Lighting will completely change the colors you use.
To pick the colors for a scene, I like to consider the environmental lighting and also the mood. If it's sad, making the color palette blue-ish will support the mood. If it's mysterious, doing darker colors with maybe a purple-ish palette will help support the mystery.
If you're having trouble picking colors for a scene, you can always color your characters with their local colors (the standard colors you choose when there aren't any other lighting factors) and then create a solid-colored layer overtop everything that is set to "Overlay" or "Soft Light." If you want everything to feel a little blue, make that Overlay layer blue, and it'll tone everything that color a little bit. You can turn the opacity down to reduce the overall effect so it's subtle, or you can have the opposite really high so it'll be more dramatic and obvious.
