Janicki and Szemplinska-Stupnicka [1], have presented approximate periodic solutions and their instabilities for low order subharmonic resonances of the twin-well potential Duffing system. They empolyed a near-linear perturbation method in their analysis. Using an improved harmonic balance technique [2, 3], the stability results we obtained are very similar, but there are some distinct differences. For example, there are differences in terms of the types and boundaries of instability, and between the solutions of the Small Orbit (SO) motion (one-well motion). Additionally, for the Large Orbit (LO) motion (cross-well motion), there is an extra saddle-type instability curve in our plot which is either not shown or is absent in reference [1]. To demonstrate these discrepancies, periodic solutions and resonance curves for both the SO and LO motions are presented. Only 1T plots are shown in our preliminary work. As verification of our solutions, a stability comparison with results obtained via numerical simulations is provided. For ease of cross-referencing, we adopted the same symbols as employed by Janicki and Szemplinska-Stupnicka in our formulation.