Tetragonal (z-in/z-out) in tetrahedral complexes?
Since $D_{h4}$ is not a subgroup of $T_d$, the z-in/z-out distortion you talk about does not give rise to a tetragonal distortion for a tetrahedral structure. It is possible to have tetragonal distortions in tetrahedral complexes, and has been observed in some cases (Bates and Chandler 1975) and (Sharnoff 1965). There is an illustration in Figure 1 of (Virot, Hayn, and Boukortt) which shows what turns out to be the easiest and most common tetragonal distortion for a tetrahedron, the one to the point group $D_{2d}$. So while z-in/z-out is associated with octohedral groups (and is usually the easiest way for them to have a tetragonal distortion), there are other distortions which occur in tetrahedrons.