In the paper [5] we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, minitwistor lines and also their moduli space, by using a double covering structure of the twistor spaces. In particular, we find that these minitwistor spaces have different properties in many respects, compared to known examples of minitwistor spaces. Especially, we show that the moduli space of the minitwistor spaces is identified with the configuration space of different 4 points on a circle divided by the standard $\mathrm{PSL}(2,\mathbf{R})$-action.