Black holes supported by self-interacting conformal scalar fields can be considered as renormalizably dressed since the conformal potential is nothing but the top power-counting renormalizable self-interaction in the relevant dimension. On the other hand, potentials defined by powers which are lower than the conformal one are also phenomenologically relevant since they are in fact super-renormalizable. In this work we provide a new map that allows one to build black holes dressed with all the (super-)renormalizable contributions starting from known conformal seeds. We explicitly construct several new examples of these solutions in dimensions D = 3 and D = 4, including not only stationary configurations but also time-dependent ones.