Efficiently computing attention maps for videos is challenging due to the motion of objects between frames. While a standard non-local search is high-quality for a window surrounding each query point, the window's small size cannot accommodate motion. Methods for long-range motion use an auxiliary network to predict the most similar key coordinates as offsets from each query location. However, accurately predicting this flow field of offsets remains challenging, even for large-scale networks. Small spatial inaccuracies significantly impact the attention module's quality. This paper proposes a search strategy that combines the quality of a non-local search with the range of predicted offsets. The method, named Shifted Non-Local Search, executes a small grid search surrounding the predicted offsets to correct small spatial errors. Our method's in-place computation consumes 10 times less memory and is over 3 times faster than previous work. Experimentally, correcting the small spatial errors improves the video frame alignment quality by over 3 dB PSNR. Our search upgrades existing space-time attention modules, which improves video denoising results by 0.30 dB PSNR for a 7.5% increase in overall runtime. We integrate our space-time attention module into a UNet-like architecture to achieve state-of-the-art results on video denoising.