  In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic
quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the
elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection
properties of (putative) $m$-ovoids of elliptic quadrics. It yields eventually
$(q+1)$-ovoids of $Q^+(7,q)$ not coming from a $1$-system. Secondly, we also
construct $m$-ovoids for $m \in \{ 2,4,6,8,10\}$ in $Q^+(7,3)$. Therefore we
first investigate how to construct spreads of $\pg(3,q)$ that have as many
secants to an elliptic quadric as possible.
