Classical scattering cross section and attractive potentials

If I want to calculate a scattering cross section $sigma$ for a classical central potential $V(r)$, the first thing to do is to obtain an expression for the angle

$$
Theta=pi-2int_{r_*(b,E)}^{infty}frac{b}{r^2sqrt{1-frac{b^2}{r^2}-V(r)/E}}dr
$$

where $r_*(b,E)$ is the turning point obtained by equating the effective potential to the total energy $E$:
$$
0=1-frac{b^2}{r_*^2}-frac{V(r_*)}{E}
$$
and $b$ stands for the impact parameter. Let’s say I want to solve this numerically. The problem is that, if the potential $V$ has ranges of values for which it is attractive, there might be choices of $(b,E)$ in which the above equation might have multiple solutions for $r_*$ (imagine for instance a Lennard-Jones type potential but two attractive wells rather than one, like below):

Which value would I choose to insert in the integral for $Theta$? Would the equation for $Theta$ change in any way? Or, how would you go about estimating $sigma$ numerically in a situation when I can have multiple solutions for $r_*$ for some choices of $(b,E)$?