Place recognition is a challenging yet crucial task in robotics. Existing 3D LiDAR place recognition methods suffer from limited feature representation capability and long search times. To address these challenges, we propose a novel coarse-to-fine framework for 3D LiDAR place recognition that combines Birds' Eye View (BEV) feature extraction, coarse-grained matching, and fine-grained verification. In the coarse stage, our framework leverages the rich contextual information contained in BEV features to produce global descriptors. Then the top-\textit{K} most similar candidates are identified via descriptor matching, which is fast but coarse-grained. In the fine stage, our overlap estimation network reuses the corresponding BEV features to predict the overlap region, enabling meticulous and precise matching. Experimental results on the KITTI odometry benchmark demonstrate that our framework achieves leading performance compared to state-of-the-art methods. Our code is available at: \url{https://github.com/fcchit/OverlapNetVLAD}.