The cyclic $6$-roots [62], [46] matrix (presented here in its dephased form) which is inequivalent to any of the remaining $6\times 6$ complex Hadamard matrices reads: $$C_6^{(0)}=\left[\begin{array}{rrrrrr} 1& 1& 1& 1& 1& 1\\ 1& -1& -d& -d^2& d^2& d\\ 1& -d^{-1}& 1& d^2& -d^3& d^2\\ 1& -d^{-2}& d^{-2}& -1& d^2& -d^2\\ 1& d^{-2}& -d^{-3}& d^{-2}& 1& -d\\ 1& d^{-1}& d^{-2}& -d^{-2}& -d^{-1}& -1\\ \end{array}\right]:\quad d=\frac{1-\sqrt{3}}{2}+i\sqrt{\frac{\sqrt{3}}{2}},$$ where $d$ is a solution of the equation $d^2+1=d\left(1-\sqrt{3}\right)$.
No affine Hadamard family stems from $C_6^{(0)}$, however, it belongs to a non-affine orbit of $B_6^{(1)}$.