No guarantees of optimality, but I'll start us off with a score of
$79!$
$\begin{matrix}9&4&5&5&0&4\\9&3&2&2&7&7\\5&3&0&1&3&8\\2&8&4&1&6&4\\9&1&7&5&2&9\\0&6&6&5&8&6\end{matrix}$
EDIT: The above answer only considers orthogonal connections (although the score is the same either way) - diagonal connections bump up the score to
$117:$
$\begin{matrix}0&0&1&1&2&0\\3&4&5&3&6&7\\7&8&9&6&4&4\\1&9&0&2&8&8\\3&0&2&5&6&0\\3&4&5&7&7&1\end{matrix}$