The nonlinear energy sink (NES) is usually coupled with a linear oscillator (LO) to rapidly transfer and immediately dissipate a significant portion of the initial shock energy induced into the LO. This passive energy transfer and dissipation are usually achieved through strong resonance captures between the NES and the LO responses. Here, a nontraditional set of nonlinear coupling restoring forces is numerically investigated to introduce enhanced versions of the NESs. In this new set of nonlinear coupling restoring forces, one has a varying nonlinear stiffness that includes both of hardening and softening stiffness components during the oscillation, which appear in closed-loops under the effect of the damping. The obtained results by the numerical simulation have shown that employing this kind of the nonlinear restoring forces for passive targeted energy transfer (TET) is promising for shock mitigation.