  In the present study, we generalize the possible ghost field configurations
within the framework of $k$-essence theory to the Simpson-Visser metric area
function $\Sigma^2=x^2+a^2$. Our analysis encompasses field configurations for
the region-defined metric function $dA_\pm$ as well as the general solution
that asymptotically behaves as Schwarzschild-de Sitter for $x\to-\infty$.
Specifically, we investigate two scalar field configurations and define the
associated potential for each one. Through rigorous calculations, we verify
that all equations of motion are satisfied. Notably, our findings indicate that
even when proposing new configurations of ghost scalar fields, the energy
conditions remain unchanged. This result serves to validate the wormhole
solutions obtained in previous studies.
