What Can I Use To Add A Custom Button Between Publish button and Move To Trash?

There is an action before the Trash link: post_submitbox_start.

You can use it to add content to that box. The Trash has a float:left, so it will move to the side.

Example:

add_action( 'post_submitbox_start', function() {
    print '<button>Hey!</button>';
});

Result: