We use the Parsons-Lee modification of Onsager's second virial theory within the restricted orientation (Zwanzig) approximation to analyze the phase behavior of hard cylindrical rods confined in narrow pores. Depending on the wall-to-wall separation we predict a number of distinctly different surface-generated nematic phases, including a biaxial planar nematic with variable number of layers, a monolayer homeotropic, and a hybrid T-type structure (a planar layer combined with a homeotropic one). For narrow pores, we find evidence of two types of second-order uniaxial-biaxial transitions depending on the aspect ratio of the particles. More specifically, we observe a continuous crossover from n to n+1 layers, each with a distinct planar anchoring symmetry as well as first-order transitions from planar to homeotropic surface anchoring. Contrary to the previously studied case of parallelepipeds we find that the surface anchoring transition from planar to homeotropic symmetry occurs at much lower overall rod packing fractions. This renders the observation of homeotropic capillary nematics much more realistic in experimental systems of strongly confined anisotropic colloids. Unlike confined parallelepipeds, cylindrical rods gradually increase the number of the nematic planar layers (without any phase transitions). However, a weakly first-order transition was observed between two planar structures with n and n+1 layers in wide pores and longer rods. In addition, the cylindrical rods exhibit a first-order transition from the homeotropic structure to the uniaxial (or biaxial) T phase that has not been observed in confined hard parallelepipeds. We further demonstrate a reentrant uniaxial-biaxial-uniaxial-biaxial phase sequence for confined cylinders at small aspect ratio. Our results also clearly demonstrate that stable T-type surface ordering is a subtle capillary effect that only becomes manifest in sufficiently narrow pores away from the two-dimensional bulk limit.