  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-B\"uttiker 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.
