Recently, a generalized successive cancellation list (SCL) decoder implemented with shifted-pruning (SP) scheme, namely the SCL-SP-ω decoder, is presented for polar codes, which is able to shift the pruning window at most ω times during each SCL re-decoding attempt to prevent the correct path from being eliminated. The candidate positions for applying the SP scheme are selected by a shifting metric based on the probability that the elimination occurs. However, the number of exponential/logarithm operations involved in the SCL-SP-ω decoder grows linearly with the number of information bits and list size, which leads to high computational complexity. In this paper, we present a detailed analysis of the SCL-SP-ω decoder in terms of the decoding performance and complexity, which unveils that the choice of the shifting metric is essential for improving the decoding performance and reducing the re-decoding attempts simultaneously. Then, we introduce a simplified metric derived from the path metric (PM) domain, and a custom-tailored deep learning (DL) network is further designed to enhance the efficiency of the proposed simplified metric. The proposed metrics are both free of transcendental functions and hence, are more hardware-friendly than the existing metrics. Simulation results show that the proposed DL-aided metric provides the best error correction performance as comparison with the state of the art.