  Given a smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geqslant
2$, we study the cones $V^h_p\subset \mathbb{P}^{n+1}$ swept out by lines
having contact order $h\geqslant 2$ at a point $p\in X$. In particular, we
prove that if $X$ is general, then for any $p\in X$ and $2 \leqslant h\leqslant
\min\{ n+1,d\}$, the cone $V^h_p$ has dimension exactly $n+2-h$. Moreover, when
$X$ is a very general hypersurface of degree $d\geqslant 2n+2$, we describe the
relation between the cones $V^h_p$ and the degree of irrationality of
$k$--dimensional subvarieties of $X$ passing through a general point of $X$. As
an application, we give some bounds on the least degree of irrationality of
$k$--dimensional subvarieties of $X$ passing through a general point of $X$,
and we prove that the connecting gonality of $X$ satisfies
$d-\left\lfloor\frac{\sqrt{16n+25}-3}{2}\right\rfloor\leqslant\conngon(X)\leqslant
d-\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor$.
