Spectral super-resolution (SSR) aims to improve the spectral resolution of images from multispectral imagery or even red, green, blue (RGB) images. However, the majority of existing SSR methods do not fully exploit the spatial and texture features in RGB images, which would lead to the image unreal and distort of the high-frequency details in the reconstructed SSR images. In this study, a hybrid spectral and texture attention pyramid network (STANet) is proposed to reconstruct hyperspectral images (HSIs) with RGB bands of remote sensing images as input. More specifically, a learnable texture feature extraction module is proposed, aiming to make full use of the texture features in the RGB images, which are important in the subsequent spectral reconstruction. Furthermore, to better reconstruct the correlations between various spectral channels, a spatial-spectral-constrained cross-attention module is introduced. Finally, a novel spectral-texture fusion method is proposed, which successfully alleviates the problem of insufficient deep interaction among multiple deep features. On three remote sensing datasets, STANet demonstrates state-of-the-art performance, with its peak signal-to-noise ratio (PSNR) exceeding the suboptimal methods by 0.7266, 0.6724, and 0.6 dB, respectively. The results of the land-cover classification experiment using the reconstructed HSI further demonstrated the performance of the STANet algorithm.