This paper proposes the use of "multicalibration" to yield interpretable and reliable confidence scores for outputs generated by large language models (LLMs). Multicalibration asks for calibration not just marginally, but simultaneously across various intersecting groupings of the data. We show how to form groupings for prompt/completion pairs that are correlated with the probability of correctness via two techniques: clustering within an embedding space, and "self-annotation" - querying the LLM by asking it various yes-or-no questions about the prompt. We also develop novel variants of multicalibration algorithms that offer performance improvements by reducing their tendency to overfit. Through systematic benchmarking across various question answering datasets and LLMs, we show how our techniques can yield confidence scores that provide substantial improvements in fine-grained measures of both calibration and accuracy compared to existing methods.