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Graph convolutional networks (GCNs) are fundamental in various scientific applications, ranging from biomedical protein-protein interactions (PPI) to large-scale recommendation systems. An essential component for modeling graph structures in GCNs is sparse general matrix-matrix multiplication (SpGEMM). As the size of graph data continues to scale up, SpGEMMs are often conducted in an out-of-core fashion due to limited GPU memory space in resource-constrained systems. Albeit recent efforts that aim to alleviate the memory constraints of out-of-core SpGEMM through either GPU feature caching, hybrid CPU-GPU memory layout, or performing the computation in sparse format, current systems suffer from both high I/O latency and GPU under-utilization issues. In this paper, we first identify the problems of existing systems, where sparse format data alignment and memory allocation are the main performance bottlenecks, and propose AIRES, a novel algorithm-system co-design solution to accelerate out-of-core SpGEMM computation for GCNs. Specifically, from the algorithm angle, AIRES proposes to alleviate the data alignment issues on the block level for matrices in sparse formats and develops a tiling algorithm to facilitate row block-wise alignment. On the system level, AIRES employs a three-phase dynamic scheduling that features a dual-way data transfer strategy utilizing a tiered memory system: integrating GPU memory, GPU Direct Storage (GDS), and host memory to reduce I/O latency and improve throughput. Evaluations show that AIRES significantly outperforms the state-of-the-art methods, achieving up to 1.8× lower latency in real-world graph processing benchmarks.