A signal picked-up technique to improve the demodulation stability and accuracy of sapphire fiber external Fabry-Perot interferometer is proposed and demonstrated. Through fusion splicing four pieces of multimode fiber in sequence with different core diameters, the in-step change of the core diameter is found to introduce a sufficient fliting effect on the transmitted higher-order guided modes in the sapphire fiber and further reduce their influence on the fundamental mode interference demodulation. Experimental results show that the proposed multi-stage coupling technique can suppress by five-fold the additional phase imposed on the fundamental mode demodulation when compared with the conventional single-stage coupling approach in which single-mode fiber is spliced with only one piece of multimode fiber. The standard deviation of the demodulated optical phase and cavity length can also be reduced by more than two times. The proposed technique provides a simple yet sufficient solution for the long-standing difficulty of multimode sapphire fiber Fabry-Perot interferometer demodulation.