When handling a large number of hard-to-predict (H2P) bra-nches, even the state-of-the-art branch predictor, TAGE-SC-L, suffers from severe table entry allocation pressure, hindering its predictive performance. Because TAGE cannot easily extract correlation from relevant history, it needs to allocate plenty of entries to memorize these branches. Using neural networks to predict these H2P branches is an effective approach. However, most existing studies are based on offline methods, these models are only effective for data similar to training data, and the expensive training and inference process makes it difficult to be practically applied in processors. To explore more practical solutions, we propose SONet, a shallow online neural network for H2P branches, with a practical training and inference architecture. At runtime, SONet identifies and selects the H2P branches, offloading suitable ones to SONet for specialized prediction, while TAGE-SC-L predicts the remaining branches. Experiments show that it improves program prediction performance where mispredictions are concentrated in a few branches. Over a set of workloads including CBP-5 and SPEC2017, a 16KB SONet backing 64KB TAGE-SC-L reduces the MPKI by 1.8%. Compared to a TAGE-SC-L of the equal capacity, our method decreases MPKI by 0.7% within acceptable prediction latency.