Consistent supercurrent multiplets are naturally associated with linearized off-shell supergravity models. In S. M. Kuzenko, J. High Energy Phys. 1004, 022 (2010) we presented the hierarchy of such supercurrents which correspond to all the models for linearized 4D N = 1 supergravity classified a few years ago. Here we analyze the correspondence between the most general supercurrent given in S. M. Kuzenko, J. High Energy Phys. 1004, 022 (2010) and the one obtained eight years ago in M. Magro et al., Ann. Phys. 298, 123 (2002) using the superfield Noether procedure. We apply the Noether procedure to the general N = 1 supersymmetric nonlinear sigma-model and show that it naturally leads to the so-called S-multiplet, revitalized in Z. Komargodski, N. Seiberg, J. High Energy Phys. 1007, 017 (2010).