We find the set of all universal minimum points of the potential of the $16$-point sharp code on $S^4$ and (more generally) of the demihypercube on $S^d$, $d\geq 5$, as well as of the $2_{41}$ polytope on $S^7$. We also extend known results on universal minima of three sharp configurations on $S^{20}$ and $S^{21}$ containing no antipodal pair to their symmetrizations about the origin. Finally, we prove certain general properties of spherical $(2m-1)$-designs contained in as few as $m$ parallel hyperplanes (all but one configuration considered here possess this property).