Self-supervised learning (SSL) has shown remarkable performance in computer vision tasks when trained offline. However, in a Continual Learning (CL) scenario where new data is introduced progressively, models still suffer from catastrophic forgetting. Retraining a model from scratch to adapt to newly generated data is time-consuming and inefficient. Previous approaches suggested re-purposing self-supervised objectives with knowledge distillation to mitigate forgetting across tasks, assuming that labels from all tasks are available during fine-tuning. In this paper, we generalize self-supervised continual learning in a practical setting where available labels can be leveraged in any step of the SSL process. With an increasing number of continual tasks, this offers more flexibility in the pre-training and fine-tuning phases. With Kaizen, we introduce a training architecture that is able to mitigate catastrophic forgetting for both the feature extractor and classifier with a carefully designed loss function. By using a set of comprehensive evaluation metrics reflecting different aspects of continual learning, we demonstrated that Kaizen significantly outperforms previous SSL models in competitive vision benchmarks, with up to 16.5% accuracy improvement on split CIFAR-100. Kaizen is able to balance the trade-off between knowledge retention and learning from new data with an end-to-end model, paving the way for practical deployment of continual learning systems.