We report that the layered semiconductors ${\mathrm{CeTe}}_{2\ensuremath{-}x}{\mathrm{Sb}}_{x}$ with low charge carrier density $\ensuremath{\le}2\ifmmode\times\else\texttimes\fi{}{10}^{18}\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ exhibit giant negative magnetoresistance exceeding two digits at 2 K. The drastic decrease in electrical resistivity is induced by the transition from the antiferromagnetic to the forced-ferromagnetic state with a wide range of electrical resistivities [approximately 20 to $(3\ifmmode\times\else\texttimes\fi{}{10}^{7})\phantom{\rule{0.28em}{0ex}}\mathrm{\ensuremath{\Omega}}\phantom{\rule{0.16em}{0ex}}\mathrm{cm}$] in zero magnetic field. In contrast, the negative magnetoresistance almost disappears in the low-resistivity sample with much higher carrier density ($\ensuremath{\sim}2.4\ifmmode\times\else\texttimes\fi{}{10}^{20}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$). These results indicate that the observed giant magnetoresistance is driven by a substantial change in the mobility of the magnetic polarons. In ${\mathrm{CeTe}}_{2\ensuremath{-}x}{\mathrm{Sb}}_{x}$, the charge carrier density is variable by several digits by the substitution of Te with Sb. Consequently, the giant negative magnetoresistance is preserved within a wide range of electrical resistivity as long as the magnetic polaron density is low enough that the magnetic polarons rarely overlap each other. The magnetic polaron diameter is discussed using the samples' carrier density.