Kim et al. (2022) studied single-machine scheduling of step-improving jobs with a common discount factor to minimize the total weighted completion time. This study proposes a pseudo-polynomial algorithm to solve the problem. We then consider the general problem in which the discount factors are job dependent. It is proven that there is an optimal solution in which the early normal jobs are sequenced in the WSPT order and the late discounted jobs are sequenced in the WSPT order. Based on two WSPT lists, a dynamic programming solution algorithm is proposed. The run time is pseudo-polynomial when the distance of any job is bounded by a constant, where the distance of a job is the absolute difference of the two positions of this job in two WSPT-ordered job lists.