Computations in anisotropic media are usually cumbersome. One particular problem occurs when two components of the slowness vector are known, yet the full slowness vector is required, e.g., to evaluate Snell's law at an interface between two anisotropic media. In this paper we suggest to combine first-order perturbation method and expressions for sectorially best-fitting isotropic background media in an iterative approach to find the third slowness component. We demonstrate the technique with examples for media with polar and triclinic symmetry.