We observe hybrid states of cavity photons and Tamm plasmons in an organic microcavity with an incorporated thin silver layer of increasing thickness up to 40 nm. Via μ-photoluminescence spectroscopy, we investigate their angular dependence. At oblique angles, we observe a TE-TM polarization splitting of more than 40 meV for each mode. An analytical model is developed to describe the coupling of Tamm plasmons and cavity photons and to account for the splitting of the orthogonally polarized resonances.