  It is known that for a smooth hyperelliptic curve to have a large $a$-number,
the genus must be small relative to the characteristic of the field, $p>0$,
over which the curve is defined. It was proven by Elkin that for a genus $g$
hyperelliptic curve $C$ to have $a_C=g-1$, the genus is bounded by
$g<\frac{3p}{2}$. In this paper, we show that this bound can be lowered to $g
<p$. The method of proof is to force the Cartier-Manin matrix to have rank one
and examine what restrictions that places on the affine equation defining the
hyperelliptic curve. We then use this bound to summarize what is known about
the existence of such curves when $p=3,5$ and $7$.
