Self-supervised learning systems have gained significant attention in recent years by leveraging clustering-based pseudo-labels to provide supervision without the need for human annotations. However, the noise in these pseudo-labels caused by the clustering methods poses a challenge to the learning process leading to degraded performance. In this work, we propose a pseudo-label refinement (SLR) algorithm to address this issue. The cluster labels from the previous epoch are projected to the current epoch cluster-labels space and a linear combination of the new label and the projected label is computed as a soft refined label containing the information from the previous epoch clusters as well as from the current epoch. In contrast to the common practice of using the maximum value as a cluster/class indicator, we employ hierarchical clustering on these soft pseudo-labels to generate refined hard-labels. This approach better utilizes the information embedded in the soft labels, outperforming the simple maximum value approach for hard label generation. The effectiveness of the proposed SLR algorithm is evaluated in the context of person re-identification (Re-ID) using unsupervised domain adaptation (UDA). Experimental results demonstrate that the modified Re-ID baseline, incorporating the SLR algorithm, achieves significantly improved mean Average Precision (mAP) performance in various UDA tasks, including real-to-synthetic, synthetic-to-real, and different real-to-real scenarios. These findings highlight the efficacy of the SLR algorithm in enhancing the performance of self-supervised learning systems.