Hyperspectral image (HSI) classification is a significant foundation for remote sensing image analysis, widely used in biology, aerospace, and other applications. Convolution neural networks (CNNs) and attention mechanisms have shown outstanding ability in HSI classification and have been widely studied in recent years. However, the existing CNN-based and attention mechanism-based methods cannot fully use spatial–spectral information, which is not conducive to further improving HSI classification accuracy. This paper proposes a new spatial–spectral Transformer network with multi-scale convolution (SS-TMNet), which can effectively extract local and global spatial–spectral information. SS-TMNet includes two key modules, i.e., multi-scale 3D convolution projection module (MSCP) and spatial–spectral attention module (SSAM). The MSCP uses multi-scale 3D convolutions with different depths to extract the fused spatial–spectral features. The spatial–spectral attention module includes three branches: height spatial attention, width spatial attention, and spectral attention, which can extract the fusion information of spatial and spectral features. The proposed SS-TMNet was tested on three widely used HSI datasets: Pavia University, IndianPines, and Houston2013. The experimental results show that the proposed SS-TMNet is superior to the existing methods.