The interaction between fluid flow and magnetic fields finds real-life applications in various industries, including semiconductors, fusion energy, plasma processing, and spacecraft propulsion. In this study, our objective is to investigate the behavior of a ternary hybrid nanofluid as it flows over a surface stretched in two directions. We consider the amalgamation of Hall current and ion slip effects, as well as homogeneous–heterogeneous reactions. The flow is influenced by Cattaneo–Christov heat flux, heat generation/absorption, and slip and convective conditions at the surface boundary. This ternary hybrid nanofluid comprises three types of nanoparticles—silicon carbide (SiC), copper oxide (CuO), and titanium oxide (TiO2)—suspended in a base fluid of diathermic oil (DO). We employ numerical methods to solve this system, utilizing the bvp4c function in MATLAB software. The results are presented graphically, demonstrating the correlation between key parameters and associated profiles. The findings reveal that the thermal profile diminishes with increasing Biot number but improves with the thermal relaxation effect. Furthermore, it is examined that the performance of the ternary hybrid nanofluid flow surpasses that of both hybrid and simple nanofluid flows. To corroborate our model, we provide a comparison with a published work in a limiting case within this study.