To overcome the challenges of weak signal demodulation in high-interference fiber-optic sensing environments, this paper proposes an improved PGC-Arctan-LHP (Phase-Generated Carrier-Arctan-Low-order Harmonic Mixing and Phase-orthogonal) algorithm. This algorithm effectively mitigates errors induced by modulation depth drift, light intensity fluctuations, and carrier phase delay. Simulations demonstrate a dynamic range of 132.06 dB at 100 Hz and a 63.09 dB signal-to-noise ratio (SNR) under composite interference. Experimental validation was performed using an all-fiber Michelson interferometric weak magnetic sensing system employing a terbium-doped silica fiber. Results demonstrate that the proposed algorithm achieves a magnetically induced phase sensitivity of 0.0092 rad/µT, thereby outperforming existing algorithms. Its noise-equivalent magnetic field sensitivity reaches 1.87 nT/√Hz at 1000 Hz, demonstrating high-precision demodulation of weak magnetic signals with significant potential for complex engineering applications.