The geology of Earth and super-Earth sized planets will, in many cases, only be observable via their atmospheres. Here, we investigate secondary volcanic atmospheres as a key base case of how atmospheres may reflect planetary geochemistry. We couple volcanic outgassing with atmospheric chemistry models to simulate the growth of C-O-H-S-N atmospheres in thermochemical equilibrium, focusing on what information about a planet's mantle fO$2$ and bulk silicate H/C ratio could be determined by atmospheric observation. 800K volcanic atmospheres develop distinct compositional groups as the mantle fO$2$ is varied, which can be identified using sets of (often minor) indicator species: Class O, representing an oxidised mantle and containing SO$2$ and sulfur allotropes; Class I, formed by intermediate mantle fO$2$'s and containing CO$2$, CH$4$, CO and COS; and Class R, produced by reduced mantles, containing H$2$, NH$3$ and CH$4$. These atmospheric classes are robust to a wide range of bulk silicate H/C ratios. However, the H/C ratio does affect the dominant atmospheric constituent, which can vary between H$2$, H$2$O and CO$2$ once the chemical composition has stabilised to a point where it no longer changes substantially with time. This final atmospheric state is dependent on the mantle fO$2$, the H/C ratio, and time since the onset of volcanism. The atmospheric classes we present are appropriate for the closed-system growth of hot exoplanets, and may be used as a simple base for future research exploring the effects of other open-system processes on secondary volcanic atmospheres.