Maximal partial line spreads of non-singular quadrics

For $n \geq 9$, we construct maximal partial line spreads for non-singular quadrics of $PG(n,q)$ for every size between approximately $(cn+d)(q^{n-3}+q^{n-5})\log{2q}$ and $q^{n-2}$, for some small constants $c$ and $d$.
These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by G\'acs and Sz\H onyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles $W_3(q)$ and $Q(4,q)$ by Pepe, R\"{o}{\ss}ing and Storme.