The link between a modified Higgs self-coupling and the strong first-order phase transition necessary for baryogenesis is well explored for polynomial extensions of the Higgs potential. We broaden this argument beyond leading polynomial expansions of the Higgs potential to higher-order polynomial terms and to non-polynomial Higgs potentials with the use of the functional renormalization group. In all cases we find that a strong first-order phase transition manifests itself in an enhancement of the Higgs self-coupling by at least 50%, implying that such modified Higgs potentials should be accessible at the LHC. The modifications of the Higgs potential can be linked to e.g. asymptotically safe extensions of the Standard Model.