Image reproduction devices, such as displays or printers, can reproduce only a limited set of colors, denoted the color gamut. The gamut depends on both theoretical and technical limitations. Reproduction device gamuts are significantly different from acquisition device gamuts. These facts raise the problem of reproducing similar color images across different devices. This is well known as the gamut mapping problem. Gamut mapping algorithms have been developed mainly using colorimetric pixel-wise principles, without considering the spatial properties of the image. The recently proposed multilevel gamut mapping approach takes spatial properties into account and has been demonstrated to outperform spatially invariant approaches. However, they have some important drawbacks. To analyze these drawbacks, we build a common framework that encompasses at least two important previous multilevel gamut mapping algorithms. Then, when the causes of the drawbacks are understood, we solve the typical problem of possible hue shifts. Next, we design appropriate operators and functions to strongly reduce both haloing and possible undesired over compression. We use challenging synthetic images, as well as real photographs, to practically show that the improvements give the expected results.