Thickness stretching is indispensable for shell formulations when local three-dimensional effects are significant. However, a thickness curvature locking phenomenon occurs and the multiplication decomposition scheme or assumed strain method is required for the conventional low-order finite element method. This work presents a 6-parameter nonlinear shell formulation with extensible thickness, implemented via a hierarchical quadrature element method (HQEM). Unlike conventional approaches requiring multiplicative director decomposition or rotational parametrization, the director field is interpolated directly as an ordinary vector in ℝ³, thereby avoiding complex treatments for finite rotations. To ensure practical applicability, a robust scheme for applying bending and twisting moments is developed. The high-order nature of HQEM inherently alleviates locking phenomena, including curvature thickness locking, without additional assumed strain or reduced integration techniques. Both Saint-Venant Kirchhoff and neo-Hookean hyperelastic models are incorporated, demonstrating the formulation's robustness and generality. Numerical examples validate the accuracy, efficiency, and locking-free performance of the method, establishing it as a reliable tool for nonlinear shell analysis.