Outliers in weights and activations pose a key challenge for fixed-point quantization of neural networks. While outliers can be addressed by fine-tuning, this is not practical for machine learning (ML) service providers (e.g., Google, Microsoft) who often receive customers' models without the training data. Specialized hardware for handling outliers can enable low-precision DNNs, but incurs nontrivial area overhead. In this paper, we propose overwrite quantization (OverQ), a novel hardware technique which opportunistically increases bitwidth for outliers by letting them overwrite adjacent values. An FPGA prototype shows OverQ can significantly improve ResNet-18 accuracy at 4 bits while incurring relatively little increase in resource utilization.