In the presence of an axion or axion-like particle (ALP) that couples to the Standard Model via dimension-five interactions, dimension-six Standard Model Effective Field Theory (SMEFT) operators are generated via the renormalization-group evolution. As many of these SMEFT contributions are experimentally tightly constrained, this "ALP-SMEFT interference" can be used to derive indirect bounds on the ALP couplings to Standard Model particles. We present a global analysis of the Wilson coefficients of the ALP effective Lagrangian based on Higgs, top, and low-energy data. The obtained bounds are model independent and are competitive with or even stronger than direct bounds in the GeV to TeV ALP-mass range.