We study a spontaneously broken Einstein–Yang–Mills–Higgs model coupled via a Higgs portal to an uncharged scalar χ. We present a phase diagram of self-gravitating solutions showing that depending on the choice of parameters of the χ scalar potential and the Higgs portal coupling constant γ, one can identify different regions: If γ is sufficiently small, a χ halo is created around the monopole core which in turn surrounds a black hole. For larger values of γ, no halo exists and the solution is just a black hole monopole one. When the horizon radius grows and becomes larger than the monopole radius, solely a black hole solution exists. Because of the presence of the χ scalar, a bound for the Higgs potential coupling constant exists and when it is not satisfied, the vacuum is unstable and no nontrivial solution exists. We briefly comment on possible connections of our results with those found in recent dark matter axion models.Instituto de Física La Plata