The thermal Hall conductivity—a thermal current analog of the Hall conductivity for electric current—is a useful detector for exotic quasiparticles emerging from quantum many-body effects, such as Majorana fermions. In particular, the half-integer quantization of the thermal Hall conductivity in the zero-temperature limit has been regarded as direct evidence of Majorana fermions in their topological Chern insulating state. However, it remains unclear how the thermal Hall conductivity behaves at finite temperatures, even in the celebrated Kitaev model on a honeycomb lattice under an external magnetic field, which is one of the fundamental models for realizing a Majorana Chern insulator in quantum magnets, despite its crucial importance in reconciling conflicting experimental observations. Here, we investigate the thermal Hall conductivity in the Kitaev model with additional interactions under a magnetic field, employing a finite-temperature tensor network method benchmarked by a thermal pure quantum state technique. We find that the thermal Hall conductivity divided by temperature, ${\kappa }_{xy}/T$, significantly overshoots the value of the half-integer quantization and exhibits a pronounced hump while decreasing temperature. Moreover, we show that the sign of ${\kappa }_{xy}/T$ for different field directions follows the weak-field perturbative prediction of the Kitaev model over a wide range of magnetic fields. We also demonstrate that the additional off-diagonal interactions, known as the $\Gamma$ and ${\Gamma }^{\prime }$ terms, considerably affect ${\kappa }_{xy}/T$. In particular, we show that positive $\Gamma$ and negative ${\Gamma }^{\prime }$ lead to a remarkable enhancement in the intermediate temperature region. From the comparison with the classical counterpart, we reveal that the effects of the $\Gamma$ term go beyond the classical picture, indicating significant quantum fluctuation effects, while those of the ${\Gamma }^{\prime }$ term are well captured at the classical level. Without assuming a specific quasiparticle picture, our approach enables a unified description of the thermal Hall response under nonperturbative magnetic fields and in the presence of additional interactions, beyond the regime directly accessible from the weak-field Majorana-based theory. This approach offers a robust framework for studying finite-temperature thermal Hall transport in Kitaev materials such as $\alpha$-RuCl${}_3$, and provides a promising pathway toward clarifying the nature of the heat-carrying excitations in correlated spin systems.