Inorganic-organic interfaces are important for enhancing the power conversion efficiency of silicon-based solar cells through singlet exciton fission (SF). We elucidated the structure of the first monolayers of tetracene (Tc), an SF molecule, on hydrogen-passivated Si(111) [H-Si(111)] and hydrogenated amorphous Si (a-Si:H) by combining near-edge X-ray absorption fine structure (NEXAFS) and Xray photoelectron spectroscopy (XPS) experiments with density functional theory (DFT) calculations. For samples grown at or below substrate temperatures of 265 K, the resulting ultrathin Tc films are dominated by almost upright-standing molecules. The molecular arrangement is very similar to the Tc bulk phase, with only a slightly higher average angle between the conjugated molecular plane normal and the surface normal (alpha) around 77 degrees. Judging from carbon K-edge X-ray absorption spectra, the orientation of the Tc molecules are almost identical when grown on H-Si(111) and a-Si:H substrates as well as for (sub)mono- to several-monolayer coverages. Annealing to room temperature, however, changes the film structure toward a smaller alpha of about 63 degrees. A detailed DFT-assisted analysis suggests that this structural transition is correlated with a lower packing density and requires a well-chosen amount of thermal energy. Therefore, we attribute the resulting structure to a distinct monolayer configuration that features less inclined, but still well-ordered molecules. The larger overlap with the substrate wave functions makes this arrangement attractive for an optimized interfacial electron transfer in SF-assisted silicon solar cells.