  This paper concerns the existence of curves with low gonality on smooth
hypersurfaces of sufficiently large degree. It has been recently proved that if
$X\subset \mathbb{P}^{n+1}$ is a hypersurface of degree $d\geq n+2$, and if
$C\subset X$ is an irreducible curve passing through a general point of $X$,
then its gonality verifies $\mathrm{gon}(C)\geq d-n$, and equality is attained
on some special hypersurfaces. We prove that if $X\subset \mathbb{P}^{n+1}$ is
a very general hypersurface of degree $d\geq 2n+2$, the least gonality of an
irreducible curve $C\subset X$ passing through a general point of $X$ is
$\mathrm{gon}(C)=d-\left\lfloor\frac{\sqrt{16n+1}-1}{2}\right\rfloor$, apart
from a series of possible exceptions, where $\mathrm{gon}(C)$ may drop by one.
