In regression settings, prediction intervals can often be too wide to directly inform decision-making, especially in situations with a low signal-to-noise ratio. In such cases, point predictions might be employed to determine actions, while prediction intervals help quantify deviations of point predictions from unseen outcomes and assess the risk associated with these decisions. Motivated by this perspective, we introduce Self-Calibrating Conformal Prediction} which builds on two post-hoc calibration approaches --- Venn-Abers calibration and conformal prediction --- to provide calibrated point predictions and associated prediction intervals with valid coverage conditional on these model predictions in finite samples. In doing so, we extend the original Venn-Abers procedure from binary classification to the regression setting. Real-data experiments illustrate how our approach improves interval efficiency by using calibrated point predictions and provides a meaningful approximation to feature-conditional validity.