I've found a better score than the existing answers of

334

As follows

$\begin{matrix}6&2&8&1&6&6\\1&8&9&3&2&8\\2&4&5&2&3&0\\3&5&4&1&7&0\\1&0&6&7&1&8\\9&9&2&5&9&2\\\end{matrix}$

Edit: Here is a new and improved answer of

337

As follows

$\begin{matrix}9&9&2&4&9&6\\1&0&6&5&1&8\\3&4&7&1&5&0\\2&7&4&2&3&0\\1&8&9&3&2&8\\0&5&8&1&6&6\\\end{matrix}$

I've found a slightly better score than the existing answers of

304

As follows

$\begin{matrix}6&2&8&1&6&6\\1&8&9&3&2&8\\3&4&5&2&3&0\\2&5&4&1&7&0\\1&0&6&7&1&8\\9&9&2&5&9&2\\\end{matrix}$

I've found a better score than the existing answers of

334

As follows

$\begin{matrix}6&2&8&1&6&6\\1&8&9&3&2&8\\2&4&5&2&3&0\\3&5&4&1&7&0\\1&0&6&7&1&8\\9&9&2&5&9&2\\\end{matrix}$