This paper investigates the problem of controlling a linear system under possibly unbounded stochastic noise with unknown convex cost functions, known as an online control problem. In contrast to the existing work, which assumes the boundedness of noise, we show that an $ \tilde{O}(\sqrt{T}) $ high-probability regret can be achieved under unbounded noise, where $ T $ denotes the time horizon. Notably, the noise is only required to have a finite fourth moment. Moreover, when the costs are strongly convex and the noise is sub-Gaussian, we establish an $ O({\rm poly} (\log T)) $ regret bound.