To make the best use of the underlying structure of faces, the collective information through face datasets and the intermediate estimates during the upsampling process, here we introduce a fully convolutional multi-stage neural network for 4$\times$× super-resolution for face images. We implicitly impose facial component-wise attention maps using a segmentation network to allow our network to focus on face-inherent patterns. Each stage of our network is composed of a stem layer, a residual backbone, and spatial upsampling layers. We recurrently apply stages to reconstruct an intermediate image, and then reuse its space-to-depth converted versions to bootstrap and enhance image quality progressively. Our experiments show that our face super-resolution method achieves quantitatively superior and perceptually pleasing results in comparison to state of the art.