Nuclear Magnetic Resonance (NMR) chemical shifts are powerful probes of local atomic and electronic structure that can be used to resolve the structures of powdered or amorphous molecular solids. Chemical shift driven structure elucidation depends critically on accurate and fast predictions of chemical shieldings, and machine learning (ML) models for shielding predictions are being increasingly used as scalable and efficient surrogates for demanding ab initio calculations. However, the prediction accuracies of current ML models still lag behind those of the DFT reference methods they approximate. Here, we introduce ShiftML3, a deep-learning model that improves the accuracy of predictions of isotropic chemical shieldings in molecular solids, and does so while also predicting the full shielding tensor. On experimental benchmark sets, we find root-mean-squared errors with respect to experiment for ShiftML3 that approach those of DFT reference calculations, with RMSEs of 0.53 ppm for 1H, 2.4 ppm for 13C, and 7.2 ppm for 15N, compared to DFT values of 0.49, 2.3, and 5.8 ppm, respectively.