For the phase demodulation of interferometric fiber optic sensors, an improved phase generated carrier (PGC) algorithm based on the linear interpolation of sampled data points was proposed to improve the dynamic range upper limit of PGC algorithms. Under a specific carrier modulation depth and initial phase conditions, data collection was performed at a frequency that was six times the carrier frequency. Through the linear interpolation of six sampling points within a single carrier cycle, six sets of orthogonal terms regarding the test signal were obtained. Phase demodulation was completed through differential cross multiplication or arctangent. Furthermore, the proposed algorithm does not require carrier mixing and low-pass filtering, thereby reducing the complexity of the algorithm. Theoretical calculation suggests that the dynamic range upper limit of the PGC algorithm can be increased by 1.58 dB. Experiment shows that when the sampling and signal frequencies are 500 kHz and 2 kHz, respectively, the peak output of the demodulation algorithm reaches 27.32 rad, upper limit of the dynamic range increases to 14.36 dB, and demodulation correlation becomes as high as 99.98%.