Recent years have witnessed the success of introducing Transformers to time series forecasting. From a data generation perspective, we illustrate that existing Transformers are susceptible to distribution shifts driven by temporal contexts, whether observed or unobserved. Such context-driven distribution shift (CDS) introduces biases in predictions within specific contexts and poses challenges for conventional training paradigm. In this paper, we introduce a universal calibration methodology for the detection and adaptation of CDS with a trained Transformer model. To this end, we propose a novel CDS detector, termed the "residual-based CDS detector" or "Reconditionor", which quantifies the model's vulnerability to CDS by evaluating the mutual information between prediction residuals and their corresponding contexts. A high Reconditionor score indicates a severe susceptibility, thereby necessitating model adaptation. In this circumstance, we put forth a straightforward yet potent adapter framework for model calibration, termed the "sample-level contextualized adapter" or "SOLID". This framework involves the curation of a contextually similar dataset to the provided test sample and the subsequent fine-tuning of the model's prediction layer with a limited number of steps. Our theoretical analysis demonstrates that this adaptation strategy is able to achieve an optimal equilibrium between bias and variance. Notably, our proposed Reconditionor and SOLID are model-agnostic and readily adaptable to a wide range of Transformers. Extensive experiments show that SOLID consistently enhances the performance of current SOTA Transformers on real-world datasets, especially on cases with substantial CDS detected by the proposed Reconditionor, thus validate the effectiveness of the calibration approach.