Standard depression assessment relies on instruments such as the clinician-rated Hamilton Depression Rating Scale (HAMD) and the patient-reported Patient Health Questionnaire (PHQ-8), but manual scoring is time-consuming and subject to inter-rater variability. Prior automated approaches typically regress a single total score or coarse severity category, lacking the fine-grained subscore-level supervision needed for precise clinical diagnosis. To address this, we propose MTSP (Multi-Task Subscore Prediction), a fine-grained model for subscore prediction via multi-task learning. MTSP achieves state-of-the-art performance on the public E-DAIC dataset (MAE 3.48, RMSE 4.57) and generalizes well to the public PDCH and a large-scale private clinical dataset (CIDH), outperforming total-score regression baselines and Qwen3-14B direct scoring. We further show that multi-task learning is essential, subscore-level supervision improves assessment by better capturing symptom-cluster structure, and prior constraints plus task-level self-paced learning enhance robustness to subscore difficulty and annotation noise.