Scheduling is important for effective production and logistics management, where tasks need to be allocated and performed by limited resources. In particular, the Job-shop Scheduling Problem (JSP) is a well-known and challenging combinatorial optimization problem in which tasks sharing a machine are to be arranged in a sequence such that encompassing jobs can be completed as early as possible. Given that for already moderately sized JSP instances, neither optimal schedules nor the runtime to termination of complete optimization methods are known, efficient approaches to approximate good-quality schedules are of interest. In this paper, we propose problem decomposition into time windows whose operations can be successively scheduled and optimized using multi-shot ASP solving. From a computational perspective, decomposition aims to split highly complex scheduling tasks into better manageable subproblems with a balanced number of operations so that good-quality or even optimal partial solutions can be reliably found in a small fraction of runtime. Regarding the feasibility and quality of solutions, problem decomposition must respect the precedence of operations within their jobs, and partial schedules optimized by time windows should yield better global solutions than obtainable in similar runtime on the full problem. We devise and investigate various decomposition strategies in terms of the number and size of time windows and heuristics for choosing their operations. Moreover, we incorporate time window overlapping and compression techniques into the iterative scheduling process to counteract the limitations of window-wise optimization restricted to partial schedules. Our experiments on JSP benchmark sets of several sizes show that successive optimization by multi-shot ASP solving leads to substantially better schedules within the runtime limit than global optimization on the full problem, where the gap increases with the number of operations to schedule.