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In this paper, we obtain the exact controllability for a refined stochastic wave equation with three controls by establishing a novel Carleman estimate for a backward hyperbolic-like operator. Compared with the known result [Q. L\"u and X Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Springer, Cham, Switzerland, 2021], the novelty of this paper is twofold: (1) Our model contains the effects in the drift terms when we put controls directly in the diffusion terms, which is more sensible for practical applications; (2) We provide an explicit description of the waiting time which is sharp in the case of dimension one and is independent of the coefficents of lower terms. [ABSTRACT FROM AUTHOR]