Narrow band gap semiconductor InSb has attracted tremendous attention due to the high electron mobility, small electron effective mass, and large Landé g-factor, which makes InSb gain potential exploitation in novel electronic and spintronic devices. In addition, with lowering dimension and downscaling size, various quantum phenomena might emerge in low-dimensional InSb structures. In this work, the magneto-transport characteristics of high-quality free-standing InSb single-crystal nanosheets are systematically explored. A large and unsaturated linear magnetoresistance is observed at temperature ranging from 125 K to 300 K. And this linear magnetoresistance exhibits non-monotonic temperature dependence, which is different with previous studies. Further analysis indicates that this phenomenon could be understood by the quantum magnetoresistance model. Thus, the study on the linear magnetoresistance of InSb is helpful to develop the physical model in InSb nanostructure and might extend to low-dimensional semiconductors. Additionally, it could lay a foundation for the future application in magnetic sensors.