Distance measurement using frequency sweeping interferometry is an absolute distance measurement technique that allows for high accuracy over long distances. Notwithstanding, the measurement accuracy is affected by laser sweeping nonlinearity and limited sweeping range. In this work, an optimized post-processing linearization method is demonstrated to realize high-accuracy arbitrary distance measurement using a laser with small modulation range. The interference signal is sparsely resampled to eliminate the influence of the sweeping nonlinearity, and the absolute distance is obtained by analyzing the phase of the resampled signal. In the measurement system, a high-finesse Fabry-Pérot cavity placed in vacuum is used as the measurement reference, so the effect of dispersion mismatch is negligible. Moreover, the distance measurement result is determined by the linear fit of the phase of each resampled point. Therefore, the influence of target vibration and other external random noise can be partially eliminated, and the reliability of the result is high. In the experiment, the sweeping range of the laser source is only 88 GHz. Comparing with a fringe-counting interferometer, the standard deviation of the residual errors is 34 µm within a distance of 6.7 m.