  In this paper we are concerned with $m$-ovoids of the symplectic polar space
${\cal W}(2n+1, q)$, $q$ even. In particular we show the existence of an
elliptic quadric of ${\rm PG}(2n+1, q)$ not polarizing to ${\cal W}(2n+1, q)$
forming a $\left(\frac{q^n-1}{q-1}\right)$-ovoid of ${\cal W}(2n+1, q)$. A
further class of $(q+1)$-ovoids of ${\cal W}(5, q)$ is exhibited. It arises by
glueing together two orbits of a subgroup of ${\rm PSp}(6, q)$ isomorphic to
${\rm PSL}(2, q^2)$. We also show that the obtained $m$-ovoids do not fall in
any of the examples known so far in the literature. Moreover, a computer
classification of the $m$-ovoids of ${\cal W}(5, 2)$ is acquired. It turns out
that ${\cal W}(5, 2)$ has $m$-ovoids if and only if $m = 3$ and that there are
exactly three pairwise non-isomorphic examples. The first example comes from an
elliptic quadric ${\cal Q}^-(5, 2)$ polarizing to ${\cal W}(5, 2)$, whereas the
other two are the $3$-ovoids previously mentioned.
