  Directly evaluated enhanced perturbative continuous unitary transformations
(deepCUTs) are used to calculate non-perturbatively extrapolated numerical data
for the ground-state energy and the energy gap. The data coincides with the
perturbative series up to the order with respect to which the deepCUT is
truncated. We develop a general scheme to extract quantum-critical properties
from the deepCUT data based on critical scaling and a strict correspondence
between the truncation used for deepCUT and the length scale of correlations at
the critical point. We apply our approach to transverse-field Ising models
(TFIMs) as paradigmatic systems for quantum phase transitions of various
universality classes depending on the lattice geometry and the choice of
antiferromagnetic or ferromagnetic coupling. In particular, we focus on the
quantum phase diagram of the bilayer antiferromagnetic TFIM on the triangular
lattice with an Ising-type interlayer coupling. Without a field, the model is
known to host a classically disordered ground state, and in the limit of
decoupled layers it exhibits the 3d-XY 'order by disorder' transition of the
corresponding single-layer model. Our starting point for the unknown parts of
the phase diagram is a high-order perturbative calculation about the limit of
isolated dimers where the model is in a gapped phase.
