In this paper, we investigated the coefficient-based regularized distribution regression for data generated by unbounded sampling processes. The algorithm adopts a two-stage sampling framework: the first-stage sample consists of probability distributions, from which the second-stage sample is drawn. A rigorous capacity-dependent convergence analysis was conducted under more general conditions, and its performance was comparable to that of one-stage sampling learning. Regularization was imposed on the coefficients and the kernel $K$ was permitted to be indefinite. The important feature of this algorithm is that it can improve the saturation effect suffered by classical kernel ridge regression (KRR). Notably, the output sample values were assumed to satisfy a moment condition (rather than the stricter uniform boundedness constraint common in related works). We derived the convergence error bounds via the novel integral operator techniques, and further established the mini-max optimal learning rates of the algorithm, which were comparable to those achieved under bounded sampling settings.