The self-force acting on a charged particle is sensitive to the global structure of curved spacetime and can serve as a probe of geometry beyond local curvature. We compute the static scalar self-force on a point charge in the two-parameter family of spherically symmetric wormholes introduced by Konoplya and Zhidenko, members of the broader Morris–Thorne class of traversable wormholes. Using mode-sum regularization, we analyze its dependence on the shape exponent $q$, which controls the throat geometry, and the redshift parameter $p$, which determines the redshift function and tidal strength. We find that the self-force is generally not unidirectional: it can change sign with radial distance from the throat, with up to two distinct zero crossings depending on $(p,q)$. This sign-switching behavior can be attributed to the competition between the wormhole’s short-range tidal gradients and the long-range effect of its topological flaring out. In the ultrastatic case $(p=0)$ where tidal forces vanish, the self-force remains strictly unidirectional. Furthermore, we provide a systematic characterization of how both the direction and large-distance falloff depend on the wormhole parameters. For sufficiently large $p$, the force can decay at a slower rate than the canonical $\sim r^{-3}$ behavior typical of isolated-body spacetimes, with stronger flaring (more negative $q$) leading to more rapid decay. In the combined limit $p\to \infty$ and $q\to -\infty$, the asymptotic falloff approaches that of the static scalar self-force in the Ellis wormhole.