  Let $K$ be a number field, $\overline{\mathbb Q}$, or the field of rational
functions on a smooth projective curve over a perfect field, and let $V$ be a
subspace of $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over
$K$ such that $V \nsubseteq Z_K$. We prove the existence of a point of small
height in $V \setminus Z_K$, providing an explicit upper bound on the height of
such a point in terms of the height of $V$ and the degree of a hypersurface
containing $Z_K$, where dependence on both is optimal. This generalizes and
improves upon the previous results of the author. As a part of our argument, we
provide a basic extension of the function field version of Siegel's lemma of J.
Thunder to an inequality with inhomogeneous heights. As a corollary of the
method, we derive an explicit lower bound for the number of algebraic integers
of bounded height in a fixed number field.
