The surface magnon polaritons (SMPs) present in thin film antiferromagnet semiconductors can be extremely confined, having most of their energy distributed within the magnetic medium. For extremely thin films these SMPs can therefore be well described using a quasimagnetostatic treatment. In this work Berreman's $4\times 4$ transfer matrix method (TMM) is used to study the SMPs supported by thicker films, beyond the magnetostatic approximation. Focus is placed on the antiferromagnet semiconductor \ce{MnF2} for which attenuated total reflection measurements are modeled, probing the hyperbolic dispersion of the medium. The dispersion relations from both the TMM and the analytical quasimagnetostatic approximation are compared. For thicker films, the coupling efficiencies into the SMP energy channels are analyzed as a function of air gap distances. The TMM is used to probe the SMP dispersion for realistic experimental configurations with modifications including type of substrate, film thickness, and crystal axis orientation. Rich phenomena are observed such as strong SMP-waveguide mode coupling and SMPs with negative refraction.