The smoothed finite element method (S-FEM), which has long been called the ``next-gen finite element method'', has already moved into the practical phase and is being implemented in commercial or general-purpose software. In the field of solid mechanics, S-FEM is classified as a strain-smoothing technique and is a pure displacement-based formulation. The SelectiveES/NS-FEM-T4 is known to be one of the best shear & volumetric locking-free S-FEM formulations using 4-node tetrahedral meshes; however, it suffers from the pressure checkerboarding issue that appears in rubber-like solids. Our group proposed a pressure-checker-free S-FEM formulation (F-barES-FEM-T4) that uses volumetric strain smoothing multiple times; yet, its computational cost is too high for practical large-scale analyses. In this study, we propose a new low-cost S-FEM formulation that achieves both locking-free and pressure-checker-free for nearly incompressible large deformation analyses. The new formulation adopts only two times volumetric strain smoothing in a different procedure from the conventional one, and successfully suppresses pressure checkerboarding to a practically acceptable level for solids with a Poisson's ratio up to 0.49.