We propose a theoretical framework to engineer hybrid-order and higher-order topological phases in three-dimensional topological insulators by coupling to $d$-wave altermagnets (AMs). The presence of only a $d_{x^2-y^2}$-type AM drives the system into a hybrid-order topological phase where both first-order and second-order topological phases coexist. This phase is characterized by spectral analysis, low-energy surface theory, and dipolar and quadrupolar winding numbers, and its signature is further confirmed by two-terminal differential conductance calculations. Incorporation of the $d_{x^2-z^2}$-type AM drives the system into two second-order topological insulator phases hosting a distinct type of hinge modes. These two variants of second-order topological phases are also topologically characterized by spectral analysis, topological invariants, low-energy surface theory, and transport calculations. Importantly, the localization and direction of propagation of these one-dimensional hinge modes are controllable by tuning the relative strengths of the altermagnetic exchange orders. We utilize this feature to propose a tunable current switching behavior mediated via the hinge modes. Our results establish an AM-based hybrid structure as a versatile platform for controllable higher-order topology and hinge-mediated device applications.

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• Magnetism
• Quantum transport
• Topological insulators
• Topological materials
• Exact diagonalization
• Landauer formula