It is known that stochastic gravitational wave backgrounds (SGWBs) have anisotropies generated by squeezed-type tensor non-Gaussianities originating from scalar-tensor-tensor (STT) and tensor-tensor-tensor cubic interactions. While the squeezed tensor non-Gaussianities in the standard slow-roll inflation with the Bunch–Davies vacuum state are suppressed due to the so-called consistency relation, those in extended models with the violation of the consistency relation can be enhanced. Among such extended models, we consider the inflation model with the non-Bunch–Davies state that is known to enhance the squeezed tensor non-Gaussianities. We explicitly formulate the primordial STT bispectrum induced during inflation in the context of Horndeski theory with the non-Bunch–Davies state and show that the induced SGWB anisotropies can be enhanced. We then discuss the detectability of those anisotropies in future gravitational wave experiments.