Remotely sensed images, especially in urban areas, have highly complex spatial distribution, since the ground objects have diverse ranges of sizes and shapes. This largely increases the difficulty of super-resolution (SR) tasks. Current deep convolutional neural network (CNN)-based SR methods often show limited performance when coping with complicated images. This article develops a second-order multi-scale super-resolution network (SMSR) to explore reconstruction tasks for difficult cases. Specifically, we propose a single-path feature reuse which cleverly captures multi-scale feature information through aggregating the features learned at different depths of a single path. Further, we present a second-order learning mechanism, which double reuses small-difference and large-difference features at local and global levels, makes use of the learned multi-scale information at maximum. The proposed methods achieve multi-scale learning using small-size convolution only, resulting in a lightweight and high-performance SR network. Experimental results show the superiority of our SMSR over state-of-the-art methods in super-resolving complicated image patterns. The effectiveness of SMSR is also demonstrated through its support to object recognition task.