  We consider the partial theta function $\theta (q,z):=\sum
_{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and
$q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $D(a):=\{ q\in \mathbb{C}$,
$0<|q|\leq a$, $\arg (q)\in [\pi /2,3\pi /2]\}$. We show that for $k\in
\mathbb{N}$ and $q\in D(0.55)$, there exists exactly one zero of $\theta (q,.)$
(which is a simple one) in the open annulus $|q|^{-k+1/2}<z<|q|^{-k-1/2}$ (if
$k\geq 2$) or in the punctured disk $0<z<|q|^{-3/2}$ (if $k=1$). For $k=1$,
$4$, $5$, $6$, $\ldots$, this holds true for $q\in D(0.6)$ as well.
