We construct and classify asymptotically flat, static, and spherically symmetric hairy black hole solutions in $U(1)$ gauge-invariant scalar-vector-tensor (SVT) theories carrying both electric and magnetic charges. Extending previous analyses restricted to ${\cal L}_{\rm SVT}^{2}$, we incorporate the cubic and quartic SVT sectors, ${\cal L}_{\rm SVT}^{3}$ and ${\cal L}_{\rm SVT}^{4}$, respectively. At the covariant level, the quartic SVT sector can generate higher-order derivatives, and we derive a condition that removes them before specializing to dyonic backgrounds, where they are generically present. We then classify the scalar hair according to the symmetry of the theory. In shift-symmetric theories, horizon regularity together with Noether-current conservation determines the scalar charge in terms of the remaining solution parameters, corresponding to secondary hair. When the couplings depend explicitly on the scalar field $\varphi$, independent scalar integration constants appear in the asymptotic solutions, allowing branches with primary hair. We also find that the magnetic charge activates the ${\tilde{f}}_3$ interaction in the cubic SVT sector, which does not contribute in purely electric static and spherically symmetric configurations, thereby producing hairy solutions supported by the magnetic charge. The scalar field also exhibits interaction-dependent asymptotic falloff rates. Combining these expansions with numerical integration, we connect the near-horizon and asymptotic regimes for all branches in the cubic sector and for one of the two quartic branches. For the remaining quartic branch, our result is restricted to the local near-horizon expansion.