Reliable reasoning in Large Language Models (LLMs) is challenged by their propensity for hallucination. While augmenting LLMs with Knowledge Graphs (KGs) improves factual accuracy, existing KG-augmented methods fail to quantify epistemic uncertainty in both the retrieved evidence and LLMs' reasoning. To bridge this gap, we introduce DoublyCal, a framework built on a novel double-calibration principle. DoublyCal employs a lightweight proxy model to first generate KG evidence alongside a calibrated evidence confidence. This calibrated supporting evidence then guides a black-box LLM, yielding final predictions that are not only more accurate but also well-calibrated, with confidence scores traceable to the uncertainty of the supporting evidence. Experiments on knowledge-intensive benchmarks show that DoublyCal significantly improves both the accuracy and confidence calibration of black-box LLMs while maintaining low token cost.