Uncertainty quantification has received considerable interest in recent works in Machine Learning. In particular, Conformal Prediction (CP) gains ground in this field. For the case of time series, Online Conformal Prediction (OCP) becomes an option to address the problem of data distribution shift over time. Indeed, the idea of OCP is to update a threshold of some quantity (whether the miscoverage level or the quantile) based on the distribution observation. To evaluate the performance of OCP methods, two key aspects are typically considered: the coverage validity and the prediction interval width minimization. Recently, new OCP methods have emerged, offering long-run coverage guarantees and producing more informative intervals. However, during the threshold update step, most of these methods focus solely on the validity of the prediction intervals~--~that is, whether the ground truth falls inside or outside the interval~--~without accounting for their relevance. In this paper, we aim to leverage this overlooked aspect. Specifically, we propose enhancing the threshold update step by replacing the binary evaluation (inside/outside) with a broader class of functions that quantify the relevance of the prediction interval using the ground truth. This approach helps prevent abrupt threshold changes, potentially resulting in narrower prediction intervals. Indeed, experimental results on real-world datasets suggest that these functions can produce tighter intervals compared to existing OCP methods while maintaining coverage validity.