Add post meta based on another post meta value before publish post

I want to add a meta value to my post , which will be based on another meta value , before publish post . This is what I have tried

add_action('save_post_my_custom_post', 'add_custom_field_automatically' );
function add_custom_field_automatically($post_ID) { 

    $new_meta_value = get_post_meta($post_ID,'_my_meta_key',TRUE).'to ' .' something new';

    add_post_meta($post_ID, '_my_new_meta_key', $new_meta_value, true);
}

But it doesn’t work . The hook fired properly , it saves only “to something new” , but what I expect the value should be “my meta value to something new”

The solution is using added_post_meta and updated_post_meta hook .

Here is the working code .

add_action( 'added_post_meta', 'add_custom_field_automatically', 10, 4 );
add_action( 'updated_post_meta', 'add_custom_field_automatically', 10, 4 );
function add_custom_field_automatically( $meta_id, $post_id, $meta_key, $meta_value )
{
    if ( '_my_meta_key' == $meta_key ) {

        add_post_meta($post_id, '_my_new_meta_key', $meta_value.'to something new', true);
    }
}