Today's HPC applications are producing extremely large amounts of data, such that data storage and analysis are becoming more challenging for scientific research. In this work, we design a new error-controlled lossy compression algorithm for large-scale scientific data. Our key contribution is significantly improving the prediction hitting rate (or prediction accuracy) for each data point based on its nearby data values along multiple dimensions. We derive a series of multilayer prediction formulas and their unified formula in the context of data compression. One serious challenge is that the data prediction has to be performed based on the preceding decompressed values during the compression in order to guarantee the error bounds, which may degrade the prediction accuracy in turn. We explore the best layer for the prediction by considering the impact of compression errors on the prediction accuracy. Moreover, we propose an adaptive error-controlled quantization encoder, which can further improve the prediction hitting rate considerably. The data size can be reduced significantly after performing the variable-length encoding because of the uneven distribution produced by our quantization encoder. We evaluate the new compressor on production scientific data sets and compare it with many other state-of-the-art compressors: GZIP, FPZIP, ZFP, SZ-1.1, and ISABELA. Experiments show that our compressor is the best in class, especially with regard to compression factors (or bit-rates) and compression errors (including RMSE, NRMSE, and PSNR). Our solution is better than the second-best solution by more than a 2x increase in the compression factor and 3.8x reduction in the normalized root mean squared error on average, with reasonable error bounds and user-desired bit-rates.