KPR preferences are generally given by
$$ \frac{c^{1-\sigma}}{1-\sigma} \cdot v(l)$$
with $l$ being leisure. Let's focus on the case where $\sigma \in (0, 1)$, where we know that $v(l)$ must be increasing and concave. Let total time be $T$, and denote $n$ by working hours. The standard problem then is
$$ \max_{n\in[0, T]} \frac{(wn)^{1-\sigma}}{1-\sigma} \cdot v(T-n) $$
An interior solution requires
$$ v(T-n) = v'(T-n)\frac{n}{1-\sigma} \tag 1$$
One increasing and concave function would be $v(x) = \frac{x^{1-\gamma}}{1-\gamma}$, $\gamma \in (0, 1)$. This yields
$$\frac{T-n}{1-\gamma} = \frac{n}{1-\sigma}$$
or
$$ n = (1-\sigma)\frac{T}{1 - \gamma + 1 - \sigma} $$
Now, generally, we should be able to pick the level of working hours - for each level of IES ($\sigma$). So let's fix $\sigma = 0$, the case of risk-neutrality, and we yield
$$ n = \frac{T}{2 - \gamma} \tag 2$$
As $\gamma \in (0, 1)$, that means that we have no fully control about the level of working hours: We cannot get that $n<0.5$ is an optimal choice with these preferences.
Note that a constant factor in $v(l)$ wouldn't help either, that would just drop out in (1). What am I missing - how can I control the level of working hours better in the multiplicative setup of KPR? That is, where $U = U(c, v(l))$ and not $U = \log c + g(v(l))$.