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The importance of building text-to-SQL parsers which can be applied to new databases has long been acknowledged, and a critical step to achieve this goal is schema linking, i.e., properly recognizing mentions of unseen columns or tables when generating SQLs. In this work, we propose a novel framework to elicit relational structures from large-scale pre-trained language models (PLMs) via a probing procedure based on Poincaré distance metric, and use the induced relations to augment current graph-based parsers for better schema linking. Compared with commonly-used rule-based methods for schema linking, we found that probing relations can robustly capture semantic correspondences, even when surface forms of mentions and entities differ. Moreover, our probing procedure is entirely unsupervised and requires no additional parameters. Extensive experiments show that our framework sets new state-of-the-art performance on three benchmarks. We empirically verify that our probing procedure can indeed find desired relational structures through qualitative analysis. Our code can be found at https://github.com/AlibabaResearch/DAMO-ConvAI.