  A small extended body moving through an external spacetime $g_{\alpha\beta}$
creates a metric perturbation $h_{\alpha\beta}$, which forces the body away
from geodesic motion in $g_{\alpha\beta}$. The foundations of this effect,
called the gravitational self-force, are now well established, but concrete
results have mostly been limited to linear order. Accurately modeling the
dynamics of compact binaries requires proceeding to nonlinear orders. To that
end, I show how to obtain the metric perturbation outside the body at all
orders in a class of generalized wave gauges. In a small buffer region
surrounding the body, the form of the perturbation can be found analytically as
an expansion for small distances $r$ from a representative worldline. Given
only a specification of the body's multipole moments, the field obtained in the
buffer region suffices to find the metric everywhere outside the body via a
numerical puncture scheme. Following this procedure at first and second order,
I calculate the field in the buffer region around an arbitrarily structured
compact body at sufficiently high order in $r$ to numerically implement a
second-order puncture scheme, including effects of the body's spin. I also
define $n$th-order (local) generalizations of the Detweiler-Whiting singular
and regular fields and show that in a certain sense, the body can be viewed as
a skeleton of multipole moments.
