In the realm of music information retrieval, similarity-based retrieval and auto-tagging serve as essential components. Given the limitations and non-scalability of human supervision signals, it becomes crucial for models to learn from alternative sources to enhance their performance. Self-supervised learning, which exclusively relies on learning signals derived from music audio data, has demonstrated its efficacy in the context of auto-tagging. In this study, we propose a model that builds on the self-supervised learning approach to address the similarity-based retrieval challenge by introducing our method of metric learning with a self-supervised auxiliary loss. Furthermore, diverging from conventional self-supervised learning methodologies, we discovered the advantages of concurrently training the model with both self-supervision and supervision signals, without freezing pre-trained models. We also found that refraining from employing augmentation during the fine-tuning phase yields better results. Our experimental results confirm that the proposed methodology enhances retrieval and tagging performance metrics in two distinct scenarios: one where human-annotated tags are consistently available for all music tracks, and another where such tags are accessible only for a subset of tracks.