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Given a smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geqslant2$, we study the cones $V^h_p\subset \mathbb{P}^{n+1}$ swept out by lineshaving contact order $h\geqslant 2$ at a point $p\in X$. In particular, weprove 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 therelation between the cones $V^h_p$ and the degree of irrationality of$k$--dimensional subvarieties of $X$ passing through a general point of $X$. Asan 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$d-\left\lfloor\frac{\sqrt{16n+25}-3}{2}\right\rfloor\leqslant\conngon(X)\leqslantd-\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor$.

Cones of lines having high contact with general hypersurfaces and applications