  We show that for every smooth generic projective hypersurface
$X\subset\mathbb P^{n+1}$, there exists a proper subvariety $Y\subsetneq X$
such that $\operatorname{codim}_X Y\ge 2$ and for every non constant
holomorphic entire map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$,
provided $\deg X\ge 2^{n^5}$. In particular, we obtain an effective
confirmation of the Kobayashi conjecture for threefolds in $\mathbb P^4$.
