Semantic segmentation by convolutional neural networks (CNN) has advanced the state of the art in pixel-level classification of remote sensing images. However, processing large images typically requires analyzing the image in small patches, and hence, features that have a large spatial extent still cause challenges in tasks, such as cloud masking. To support a wider scale of spatial features while simultaneously reducing computational requirements for large satellite images, we propose an architecture of two cascaded CNN model components successively processing undersampled and full-resolution images. The first component distinguishes between patches in the inner cloud area from patches at the cloud's boundary region. For the cloud-ambiguous edge patches requiring further segmentation, the framework then delegates computation to a fine-grained model component. We apply the architecture to a cloud detection data set of complete Sentinel-2 multispectral images, approximately annotated for minimal false negatives in a land-use application. On this specific task and data, we achieve a 16% relative improvement in pixel accuracy over a CNN baseline based on patching.