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For the combinatorial graph alignment problem (GAP), finding the node correspondence that maximizes the number of common edges (nce) between two unlabeled graphs, the classical Frank-Wolfe heuristic FAQ remains a strong baseline once it is properly initialized, while existing GNN approaches struggle in the purely structural setting. We introduce a chaining procedure: a sequence of Folklore-type (2-FWL) GNNs in which each network is trained with cross-entropy after decoding the previous network's similarity matrix and ranking nodes by their current alignment quality. This non-differentiable ranking step injects discrete combinatorial feedback at every stage; at inference, we iterate the final network and keep the candidate with highest observed nce. On sparse Erdos-Renyi graphs at noise level 0.25, chained FGNNs used to initialize FAQ reach 85% accuracy versus 13% for FAQ initialized from the convex relaxation, and essentially 0% for prior GNN methods. On correlated regular graphs, where MPNNs with constant features produce identical node embeddings (1-WL fails to refine) and FAQ's convex initialization is degenerate, chaining is the only method among those we evaluate, including Gromov-Wasserstein solvers, that recovers a non-trivial alignment. On three real-world benchmarks (yeast PPI, coauthorship, and road networks), recent comparisons underestimate FAQ by initializing it from a uniform doubly stochastic matrix; initialized from the convex relaxation it already surpasses prior reported numbers, and dataset-specific chained FGNNs further improve on this strengthened baseline. Code and pretrained chains available at: https://github.com/mlelarge/chaining-gnn-graph-alignment.