Frequency scanning interferometry using state-of-the-art high-speed frequency-swept laser source can be utilized to measure absolute distance on the order of micrometers to centimeters. Current distance demodulation methods based on fast Fourier transform (FFT) or fringe counting cannot achieve satisfactory accuracy when the number of sampling points within a frequency-sweeping period is small; the conventional Hilbert transform is more accurate, but it needs arctangent calculation and phase unwrapping, which is time consuming. So we propose a fast algorithm based on the conventional Hilbert transform to recover the distance from the interference signal. The algorithm is implemented by first performing a Hilbert transform and then solving the phase and the distance from the Hilbert signal with a novel, to the best of our knowledge, method that eliminates the need for arctangent calculation and phase unwrapping. The whole process took only 40 µs, and it is almost 2 times faster than the conventional Hilbert algorithm with little accuracy lost. Simulation results demonstrate that the proposed algorithm is more accurate than the FFT algorithm, and it achieved a standard deviation of 0.062 µm, which was less than that of the FFT, in our experiment at a distance of approximately 16 mm and measurement speed of 1 kHz.