The hypergroups $H$ of type $U$ on the right can be classified in terms of the family $P_{1}=\{1\circ x\mid x\in H\}$, where $1\in H$ is the right scalar identity. If the size of $H$ is $5$, then $P_{1}$ can assume only $6$ possible values, three of which have been studied inthe first part of the paper. In this paper, we completely describe other two of the remaining possible cases: a)~$P_{1}=\{\{1\},\{2,3\},\{4\},\{5\}\}$; b)~$P_{1}=\{\{1\},\{2,3\},\{4,5\} }$. In these cases, $P_{1}$ is a partition of $H$ and the equivalence relation associated to it is a regular equivalence on $H$. We find that, apart of isomorphisms, there are exactly $41$ hypergroups in case~a), and $56$ hypergroup in case~b).