  Motivated by the lack of a general parametrization for exotic compact
objects, we construct a class of perturbative solutions valid for small (but
otherwise generic) multipolar deviations from a Schwarzschild metric in general
relativity. We introduce two classes of exotic compact objects, with "soft" and
"hard" hair, for which the curvature at the surface is respectively comparable
to or much larger than that at the corresponding black-hole horizon. We extend
the Hartle-Thorne formalism to relax the assumption of equatorial symmetry and
to include deformations induced by multipole moments higher than the spin, thus
constructing the most general, axisymmetric quasi-Schwarzschild solution to
Einstein's vacuum equations. We explicitly construct several particular
solutions of objects with soft hair, which might be useful for tests of
quasi-black-hole metrics, and also to study deformed neutron stars. We show
that the more compact a soft exotic object is, the less hairy it will be. All
its multipole moments can approach their corresponding Kerr values only in two
ways as their compactness increases: either logarithmically (or faster) if the
moments are spin-induced, or linearly (or faster) otherwise. Our results
suggest that it is challenging (but possibly feasible with next-generation
gravitational-wave detectors) to distinguish Kerr black holes from a large
class of ultracompact exotic objects on the basis of their different multipolar
structure.
