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Abstract As climate change advances, environmental gradients may decouple, generating novel multivariate environments that stress wild populations. A commonly invoked mechanism of evolutionary rescue is adaptive gene flow tracking climate shifts, but gene flow from populations inhabiting similar conditions on one environmental axis could cause maladaptive introgression when populations are adapted to different environmental variables that do not shift together. Genomic architecture can play an important role in determining the effectiveness and relative magnitudes of adaptive gene flow and in situ adaptation. This may have direct consequences for how species respond to climate change but is often overlooked. Here, we simulated microevolutionary responses to environmental change under scenarios defined by variation in the polygenicity, linkage, and genetic redundancy of two independent traits, one of which is adapted to a gradient that shifts under climate change. We used these simulations to examine how genomic architecture influences evolutionary outcomes under climate change. We found that climate‐tracking (up‐gradient) gene flow, though present in all scenarios, was strongly constrained under scenarios of lower linkage and higher polygenicity and redundancy, suggesting in situ adaptation as the predominant mechanism of evolutionary rescue under these conditions. We also found that high polygenicity caused increased maladaptation and demographic decline, a concerning result given that many climate‐adapted traits may be polygenic. Finally, in scenarios with high redundancy, we observed increased adaptive capacity. This finding adds to the growing recognition of the importance of redundancy in mediating in situ adaptive capacity and suggests opportunities for better understanding the climatic vulnerability of real populations.