© 2014 American Physical Society. We show that a particular class of postrecombination phase transitions in the vacuum can lead to localized overdense regions on relatively small scales, roughly 106 to 1010M⊙, potentially interesting for the origin of large black hole seeds and for dwarf galaxy evolution. Our study suggests that this mechanism could operate over a range of conditions which are consistent with current cosmological and laboratory bounds. One byproduct of phase transition bubble-wall decay may be extra radiation energy density. This could provide an avenue for constraint, but it could also help reconcile the discordant values of the present Hubble parameter (H0) and σ8 obtained by cosmic microwave background (CMB) fits and direct observational estimates. We also suggest ways in which future probes, including CMB considerations (e.g., early dark energy limits), 21-cm observations, and gravitational radiation limits, could provide more stringent constraints on this mechanism and the sub-eV scale beyond-standard-model physics, perhaps in the neutrino sector, on which it could be based. Late phase transitions associated with sterile neutrino mass and mixing may provide a way to reconcile cosmological limits and laboratory data, should a future disagreement arise.