Average Nodal Pressure (ANP) is a simple and useful technique to alleviate the volumetric locking for all element types of standard FEM, including linear 3-node Triangles (T3) and 4-node Tetrahedrons (T4). However, standard FEM using T3 and T4 elements has shown interior accuracy and convergence than FS-FEM using same elements in previous literatures. In this paper, we combine FS-FEM and ANP to propose FS-FEM/ANP using linear T4 element for nearly-incompressible solids. The proposed FS-FEM/ANP-T4 is used to calculate a benchmark, 3D Lame problem. This 3D Lame benchmark proves that FS-FEM/ANP-T4 is free of volumetric locking, more accurate and converging faster than FEM/ANP-T4. Meanwhile, FS-FEM/ANP-T4 still possesses the remarkable endurance of mesh distortion. Also, a rubber beam applied with pressure is calculated to verify the good stability of FS-FEM/ANP-T4 on large deformation. In addition, proposed FS-FEM/ANP-T4 is used to simulate an application, a rubber hanger loaded with exhaust gravity. Comparisons in these examples with analytical results and other methods results show FS-FEM/ANP-T4 is a better alternative of FEM/ANP-T4.