The development of a multi-cut stochastic approximation (SA) method for solving stochastic convex composite optimization (SCCO) problems has remained an open challenge. The difficulty arises from the fact that the stochastic multi-cut model, constructed as the pointwise maximum of individual stochastic linearizations, provides a biased estimate of the objective function, with the error being uncontrollable. This paper introduces multi-cut SA methods for solving SCCO problems, achieving near-optimal convergence rates. The cutting-plane models used in these methods are the pointwise maxima of appropriately chosen one-cut models. To the best of our knowledge, these are the first multi-cut SA methods specifically designed for SCCO problems. Finally, computational experiments demonstrate that these methods generally outperform both the robust stochastic approximation method and the stochastic dual averaging method across all instances tested.