Among hyperstructures of type U on the right having small size, the order 6 is a relevant case. Indeed, only if the order is \leq 6 there exist proper semihypergrops and hypergroups of type U on the right whose right scalar identity is not also left identity. In the present paper we show a construction of hypergroups of type U on the right whose right scalar identity is not also left identity. That construction characterizes completely the case of order 6, and allows to introduce a semi-ordering structure within that case. With the help of that semi-ordering, and of symbolic computation software, we show that these hypergroups can be obtained as hyperproduct extensions of 41 minimal hypergroups, and that the number of their isomorphism classes is 946.
