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The smoothed finite element method (S-FEM) [1] has long been called the “next-gen finiteelement method” and is now in the practical phase. The edge-based S-FEM using 4-nodetetrahedral meshes (ES-FEM-T4) and the combined method with the edge- and node-basedformulations (SelectiveES/NS-FEM-T4) have already been implemented in commercial andgeneral-purpose software [2, 3], respectively.For the application of S-FEM-T4s to nearly incompressible solids, however, the pressure checker-boarding issue remains unresolved to this day. SelectiveES/NS-FEM-T4 [4, 5] is locking-freebut not pressure-checker-free; thus, it is not satisfactory for the analyses of rubber-like solids,biomaterials, metals under plastic deformation, polymers under heat process, and so on. To sup-press pressure checkerboarding, our group has proposed several improved S-FEM-T4s, such asF-barES-FEM-T4(c) [6], but they are relatively high-cost and thus inadequate for practical use.In this study, we propose a new S-FEM-T4 formulation (ES-FEM-SRI-T4) to achieve low-cost, locking-free, and pressure-checker-free in nearly incompressible large deformation anal-yses. ES-FEM-SRI-T4 adopts the classical edge-based one (ES-FEM-T4) for the deviatoricpart, whereas it adopts a novel round-trip NS-FEM-T4 with inconsistent stress integration (RT-NS-FEM-T4) for the volumetric part. The deviatoric and volumetric parts are combined withthe selective reduced integration (SRI) method in the same fashion as SelectiveES/NS-FEM-T4. ES-FEM-T4 for the deviatoric part resolves the shear locking issue, whereas RT-NS-FEM-T4 for the volumetric part resolves volumetric locking and pressure checkerboarding issue forsolids with a Poisson’s ratio up to 0.49. The computational cost of ES-FEM-SRI-T4 is the sameas that of SelectiveES/NS-FEM-T4, which is fast enough for practical use.