We consider the problem of training a deep neural network on a given classification task, e.g., ImageNet-1K (IN1K), so that it excels at that task as well as at other (future) transfer tasks. These two seemingly contradictory properties impose a trade-off between improving the model's generalization while maintaining its performance on the original task. Models trained with self-supervised learning (SSL) tend to generalize better than their supervised counterparts for transfer learning; yet, they still lag behind supervised models on IN1K. In this paper, we propose a supervised learning setup that leverages the best of both worlds. We enrich the common supervised training framework using two key components of recent SSL models: multi-scale crops for data augmentation and the use of an expendable projector head. We replace the last layer of class weights with class prototypes computed on the fly using a memory bank. We show that these three improvements lead to a more favorable trade-off between the IN1K training task and 13 transfer tasks. Over all the explored configurations, we single out two models: t-ReX that achieves a new state of the art for transfer learning and outperforms top methods such as DINO and PAWS on IN1K, and t-ReX* that matches the highly optimized RSB-A1 model on IN1K while performing better on transfer tasks. Project page and pretrained models: https://europe.naverlabs.com/t-rex.