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Chi et al. recently proposed two effective non-cancellation multistage (NCMS) blind source separation algorithms, one using the turbo source extraction algorithm (TSEA), called the NCMS-TSEA, and the other using the fast kurtosis maximization algorithm (FKMA), called the NCMS-FKMA. Their computational complexity and performance heavily depend on the dimension of multi-sensor data, i.e., number of sensors. This paper proposes the inclusion of the prewhitening processing in the NCMS-TSEA and NCMS-FKMA before performing source extraction. We come up with two improved algorithms with significant computational savings on one hand, and some performance improvements on the other hand (owing to dimension reduction and noise reduction by prewhitening processing), especially when the number of sensors is much larger than the number of sources. Simulation results are presented to verify the efficacy and computational efficiency of the proposed algorithms.