  We show that reconstructing a curve in $\mathbb{R}^d$ for $d\geq 2$ from a
$0.66$-sample is always possible using an algorithm similar to the classical
NN-Crust algorithm. Previously, this was only known to be possible for
$0.47$-samples in $\mathbb{R}^2$ and $\frac{1}{3}$-samples in $\mathbb{R}^d$
for $d\geq 3$. In addition, we show that there is not always a unique way to
reconstruct a curve from a $0.72$-sample; this was previously only known for
$1$-samples. We also extend this non-uniqueness result to hypersurfaces in all
higher dimensions.
