Seam carving and its variants are popular as content-aware image resizing methods. However, they often suffer from the problem that excessive downscaling causes perceptually annoying distortions. This is mainly because penetration of the seams into some important objects becomes unavoidable at the latter stage of the processing. As a solution for this problem, we previously proposed a nonlinear downscaling technique which iteratively performed a DCT-based locally linear scaling operator within 'belt-like seams', i.e. seams with a certain width. To enhance this idea, in this paper, we replace the latter processing stage with a global linear scaling operator. A transition point between the nonlinear and linear processing stages is automatically determined based on a preservation measurement for the important objects. Simulation results show that our approach can produce subjectively better results than the conventional nonlinear downscaling methods.