Many industrial and scientific applications require optimization of one or more objectives by tuning dozens or hundreds of input parameters. While Bayesian optimization has been a popular approach for the efficient optimization of blackbox functions, its performance decreases drastically as the dimensionality of the search space increases (i.e., above twenty). Recent advancements in high-dimensional Bayesian optimization (HDBO) seek to mitigate this issue through techniques such as adaptive local search with trust regions or dimensionality reduction using random embeddings. In this paper, we provide a close examination of these advancements and show that sampling strategy plays a prominent role and is key to tackling the curse-of-dimensionality. We then propose cylindrical Thompson sampling (CTS), a novel strategy that can be integrated into single- and multi-objective HDBO algorithms. We demonstrate this by integrating CTS as a modular component in state-of-the-art HDBO algorithms. We verify the effectiveness of CTS on both synthetic and real-world high-dimensional problems, and show that CTS largely enhances existing HDBO methods.