We propose a simple model that can alleviate the H0 tension while remaining consistent with big bang nucleosynthesis (BBN). It is based on a dark sector described by a standard Lagrangian featuring a SU(N) gauge symmetry with N≥3 and a massive scalar field with a quartic coupling. The scalar acts as a dark Higgs leading to spontaneous symmetry breaking SU(N)→SU(N-1) via a first-order phase transition à la Coleman-Weinberg. This setup naturally realizes previously proposed scenarios featuring strongly interacting dark radiation (SIDR) with a mass threshold within hot new early dark energy. For a wide range of reasonable model parameters, the phase transition occurs between the BBN and recombination epochs and releases a sufficient amount of latent heat such that the model easily respects bounds on extra radiation during BBN while featuring a sufficient SIDR density around recombination for increasing the value of H0 inferred from the cosmic microwave background. Our model can be summarized as a natural mechanism providing two successive increases in the effective number of relativistic degrees of freedom after BBN but before recombination ΔNBBN→ΔNNEDE→ΔNIR alleviating the Hubble tension. The first step is related to the phase transition, and the second is related to the dark Higgs becoming nonrelativistic. This setup predicts further signatures, including a stochastic gravitational wave background and features in the matter power spectrum that can be searched for with future pulsar timing and Lyman-α forest measurements.