  We propose a simple model that can alleviate the $H_0$ 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 $N\geq3$ and a massive scalar field with a quartic coupling. The scalar
acts as dark Higgs leading to spontaneous symmetry breaking $SU(N)\to
SU(N\!-\!1)$ via a first-order phase transition \`a la Coleman-Weinberg. This
set-up naturally realizes previously proposed scenarios featuring strongly
interacting dark radiation (SIDR) with a mass threshold within hot new early
dark energy (NEDE). 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 $H_0$ 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 $\Delta N_\mathrm{BBN}
\to \Delta N_\mathrm{NEDE} \to \Delta N_\mathrm{IR}$ alleviating the Hubble
tension. The first step is related to the phase transition and the second to
the dark Higgs becoming non-relativistic. This set-up 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-$\alpha$ forest measurements.
