The progressive integration of cyber-physical systems in smart grids raises potential security concerns, exacerbating the risk of false data injection attack (FDIA) that leads to severe operational disruptions, especially when the FDIA displays profiles that deviate from known attack patterns. Current FDIA detection methods usually operate under the assumption of distributional consistency between training and testing data, thereby falling short in recognizing FDIA with such out-of-distribution (OOD) characteristics. To address this challenge, this paper proposes a novel logit-normalized Bayeisan ResNet (LNBRN) algorithm, a cutting-edge method to address the unexplored OOD FDIA issues efficiently. During offline training, the proposed LNBRN leverages dropout techniques to approximate Bayesian variational inference, thus reducing computational overhead. A key point is the introduction of logit normalization to the output layer, which significantly alleviate the model overconfidence and enhance the follow-up OOD detection performance. During online detection, LNBRN incorporates mutual information to quantify the epistemic uncertainty for incoming measurements, enabling accurate identification of high-risk OOD FDIA events. Comprehensive experimental evaluations on IEEE 14-bus and 118-bus test systems with real load data demonstrate the superiority of detecting OOD FDIA and validate the scalability in larger smart grids.