We demonstrate an enhanced, bidirectional, in-plane magnetic field tuning of the gyrotropic resonance frequency of a magnetic vortex within a disk by introducing a flat edge. When the core is in its vicinity, the flat edge locally reduces the core's directional dynamic stiffness for movement parallel to the edge. This strongly reduces the net dynamic core stiffness, leading to the gyrotropic frequency being significantly less than when the core is centered (or located near the round edge). This leads to the measurable range of gyrotropic frequencies being more than doubled and also results in a clear chirality-mediated bistability of the gyrotropic resonance frequency due to what is effectively a chirality-dependence of the core's confining potential.