  We apply a new method with explicit solution operators to construct
asymptotically flat initial data sets of the vacuum Einstein equation with new
localization properties. Applications include an improvement of the decay rate
in Carlotto--Schoen [arXiv:1407.4766] to $\mathcal{O}(|x|^{-(d-2)})$ and a
construction of nontrivial asymptotically flat initial data supported in a
degenerate sector $\{(x',x_d)\in\mathbb{R}^d:|x'|\leq x_d^\alpha\}$ for
$\frac{3}{d+1}<\alpha<1$.
