The contextual multi-armed bandit (MAB) problem is crucial in sequential decision-making. A line of research, known as online clustering of bandits, extends contextual MAB by grouping similar users into clusters, utilizing shared features to improve learning efficiency. However, existing algorithms, which rely on the upper confidence bound (UCB) strategy, struggle to gather adequate statistical information to accurately identify unknown user clusters. As a result, their theoretical analyses require several strong assumptions about the "diversity" of contexts generated by the environment, leading to impractical settings, complicated analyses, and poor practical performance. Removing these assumptions has been a long-standing open problem in the clustering of bandits literature. In this work, we provide two partial solutions. First, we introduce an additional exploration phase to accelerate the identification of clusters. We integrate this general strategy into both graph-based and set-based algorithms and propose two new algorithms, UniCLUB and UniSCLUB. Remarkably, our algorithms require substantially weaker assumptions and simpler theoretical analyses while achieving superior cumulative regret compared to previous studies. Second, inspired by the smoothed analysis framework, we propose a more practical setting that eliminates the requirement for i.i.d. context generation used in previous studies, thus enhancing the performance of existing algorithms for online clustering of bandits. Extensive evaluations on both synthetic and real-world datasets demonstrate that our proposed algorithms outperform existing approaches.