  We discuss how the assumption of variable equation of state (EoS) allows the
elimination of the instability at equilibrium throat radius $a_0=3M$ featured
by previous Schwarzschild thin-shell wormhole models. Unobstructed stability
regions are found for three choices of variable EoS. Two of these EoS entail
linear stability at every equilibrium radius. Particularly, the thin-shell
remains stable as $a_0$ approaches the Schwarzschild radius $2M$. A
perturbative analysis of the wormhole equation of motion is carried out in the
case of variable Chaplygin EoS. The squared proper angular frequency
$\omega_0^2$ of small throat oscillations is linked with the second derivative
of the thin-shell potential. In various situations $\omega_0^2$ remains
positive and bounded in the limit $a_0\rightarrow 2M$.
