  We show the Kontsevich space of rational curves of degree at most roughly
$\frac{2-\sqrt{2}}{2}n$ on a general hypersurface $X\subset \mathbb{P}^n$ of
degree $n-1$ is equidimensional of expected dimension and has two components:
one consisting generically of smooth, embedded rational curves and the other
consisting of multiple covers of a line. This proves more cases of a conjecture
of Coskun, Harris, and Starr and shows the Gromov-Witten invariants in these
cases are enumerative.
