Speech bandwidth extension (BWE) aims to reconstruct high-fidelity wideband audio from narrowband inputs. While recent approaches have made significant progress, they often struggle to reconstruct realistic high-frequency phase and harmonic structures, leading to perceptual artifacts. In this paper, we propose FSC-Net (Full-Spectrum Context Network), a parameter-efficient architecture designed to explicitly model cross-band harmonic dependencies. By integrating Fast Fourier Convolutions (FFCs) into a complex spectral mapping framework, FSC-Net expands its receptive field to the entire spectrum, capturing long-range frequency interactions effectively. To address the ill-posed nature of high-frequency generation, our novel frequency-progressive learning curriculum guides the network to reconstruct spectral details from coarse to fine. Experimental results on the VCTK and unseen EARS datasets demonstrate that FSC-Net delivers consistently strong reconstruction quality and generalization, particularly in the challenging VCTK 4 kHz-to-48 kHz task. Compared to scaled-up baselines, our model attains leading LSD and PESQ scores while maintaining a highly compact parameter footprint (1.54 M).