Quantization of deep neural networks (DNN) has become a key element in the efforts of embedding such networks on end-user devices. However, current quantization methods usually suffer from costly accuracy degradation. In this paper, we propose a new method for Enhanced Post Training Quantization named EPTQ. The method is based on knowledge distillation with an adaptive weighting of layers. In addition, we introduce a new label-free technique for approximating the Hessian trace of the task loss, named Label-Free Hessian. This technique removes the requirement of a labeled dataset for computing the Hessian. The adaptive knowledge distillation uses the Label-Free Hessian technique to give greater attention to the sensitive parts of the model while performing the optimization. Empirically, by employing EPTQ we achieve state-of-the-art results on a wide variety of models, tasks, and datasets, including ImageNet classification, COCO object detection, and Pascal-VOC for semantic segmentation. We demonstrate the performance and compatibility of EPTQ on an extended set of architectures, including CNNs, Transformers, hybrid, and MLP-only models.