We study the spectral minimal $k$-partitions of the two-dimensional flat torus $\left(\RR/\ZZ\right)\times\left(\RR/b\ZZ\right)\,$, with $b$ a parameter in $b\in (0,1]$. We give a heuristic argument to compute a transition value when $k$ is odd. We support this conjecture by looking for candidates to be minimal partitions using an optimization algorithm adapted from \cite{BouBucOud09}. Guided by these numerical results, we construct $k$-partitions that are tilings of the torus by hexagons. We compute their energy and thus obtain an upper bound of the minimal energy.
