In this paper we propose and analyse a large margin fine-tuning strategy and a quality-aware score calibration in text-independent speaker verification. Large margin fine-tuning is a secondary training stage for DNN based speaker verification systems trained with margin-based loss functions. It enables the network to create more robust speaker embeddings by enabling the use of longer training utterances in combination with a more aggressive margin penalty. Score calibration is a common practice in speaker verification systems to map output scores to well-calibrated log-likelihood-ratios, which can be converted to interpretable probabilities. By including quality features in the calibration system, the decision thresholds of the evaluation metrics become quality-dependent and more consistent across varying trial conditions. Applying both enhancements on the ECAPA-TDNN architecture leads to state-of-the-art results on all publicly available VoxCeleb1 test sets and contributed to our winning submissions in the supervised verification tracks of the VoxCeleb Speaker Recognition Challenge 2020.