A non-affine family of complex Hadamard matrices found by M. Matolcsi and F. Szöllősi reads $$M_6^{(1)}(a)=\left[\begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -1 & a & a & -a & -a\\ 1 & a & b & c & d & e\\ 1 & a & c & b & e & d\\ 1 & -a & d & e & f & g\\ 1 & -a & e & d & g & f \end{array}\right]:\quad a=\exp(it): t\in\left(\frac{\pi}{2},\pi\right]\cup\left(\pi\frac{3}{2},2\pi\right]$$, where $$\begin{aligned} b(a) &= \frac{1}{4}\frac{1-6a^2+a^4-\sqrt{1+4a^2-58a^4+4a^6+a^8}}{a^2+2a-1},\\ c(a) &= \frac{1}{4}\frac{1-6a^2+a^4+\sqrt{1+4a^2-58a^4+4a^6+a^8}}{a^2+2a-1},\\ d(a) &= -\frac{1+a^2}{4}-i\frac{1}{4}\frac{1+a^2}{|1+a^2|}\sqrt{16-|1+a^2|^2},\\ e(a) &= -\frac{1+a^2}{4}+i\frac{1}{4}\frac{1+a^2}{|1+a^2|}\sqrt{16-|1+a^2|^2},\\ f(a) &= \frac{a^2+2a-1}{4}+i\frac{1}{4}\frac{a^2+2a-1}{|a^2+2a-1|}\sqrt{16-|a^2+2a-1|^2},\\ g(a) &= \frac{a^2+2a-1}{4}-i\frac{1}{4}\frac{a^2+2a-1}{|a^2+2a-1|}\sqrt{16-|a^2+2a-1|^2}. \end{aligned}$$
Family $M_6^{(1)}$ connects $F_6^{(2)}$ and $D_6^{(1)}$ $:M_6^{(1)}\simeq F_6$. Moreover, taking a particular sequence, one recovers $\lim_{t\to\pi/2}M_6\left(e^{it}\right)\simeq D_6$ for $t\in\left(-\frac{\pi}{2},2\pi\right]$. This implies that the set of known (up to November of 2008) complex Hadamard matrices of order $6$ is connected, except from Tao's isolated matrix $S_6^{(0)}$.