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We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the -norm if their initial data belong to a finite dimensional subspace . In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate when or when is even and a specific condition is satisfied, provided that their initial data is sufficiently small. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. As a consequence, we rediscover known results which were previously established for or for the smallest critical length with by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than for all critical lengths.