  We consider super processes whose spatial motion is the $d$-dimensional
Brownian motion and whose branching mechanism $\psi$ is critical or
subcritical; such processes are called $\psi$-super Brownian motions. If
$d\!>\!2\bgamma/(\bgamma\!-\!1)$, where $\bgamma\!\in\!(1,2]$ is the lower
index of $\psi$ at $\infty$, then the total range of the $\psi$-super Brownian
motion has an exact packing measure whose gauge function is $g(r)\! =\!
(\log\log1/r) / \varphi^{-1} ( (1/r\log\log 1/r)^{2})$, where $\varphi\! =\!
\psi^\prime\! \circ \! \psi^{\!-1}$. More precisely, we show that the
occupation measure of the $\psi$-super Brownian motion is the $g$-packing
measure restricted to its total range, up to a deterministic multiplicative
constant only depending on $d$ and $\psi$. This generalizes the main result of
\cite{Duq09} that treats the quadratic branching case. For a wide class of
$\psi$, the constant $2\bgamma/(\bgamma\!-\!1)$ is shown to be equal to the
packing dimension of the total range.
