We investigate a generalized multi-orbital tight-binding model on a triangular lattice, a system prevalent in a wide range of two-dimensional materials, and particularly relevant for simulating transition metal dichalcogenide monolayers. We show that the interplay between spin-orbit coupling and different symmetry-breaking mechanisms leads to the emergence of four distinct topological phases [Eck, P., \textit{et al.}, Phys. Rev. B, 107 (11), 115130 (2023)]. Remarkably, this interplay also triggers the orbital Hall effect with distinguished characteristics. Furthermore, by employing the Landauer-Büttiker formula, we establish that in the orbital Hall insulating phase, the orbital angular momentum is carried by edge states present in nanoribbons with specific terminations. We also show that, as expected, they do not have topological protection against the disorder of the edge states belonging to a first-order topological insulator.