Thompson Sampling provides an efficient technique to introduce prior knowledge in the multi-armed bandit problem, along with providing remarkable empirical performance. In this paper, we revisit the Thompson Sampling algorithm under rewards drawn from symmetric alpha-stable distributions, which are a class of heavy-tailed probability distributions utilized in finance and economics, in problems such as modeling stock prices and human behavior. We present an efficient framework for posterior inference, which leads to two algorithms for Thompson Sampling in this setting. We prove finite-time regret bounds for both algorithms, and demonstrate through a series of experiments the stronger performance of Thompson Sampling in this setting. With our results, we provide an exposition of symmetric alpha-stable distributions in sequential decision-making, and enable sequential Bayesian inference in applications from diverse fields in finance and complex systems that operate on heavy-tailed features.