Editing edit-tags.php page in wp-admin

You can use the dynamic {$taxonomy}_term_new_form_tag action, where you replace {$taxonomy} with your taxonomy slug.

Example¹⁾

For the post_tag taxonomy:

add_action( 'post_tag_term_new_form_tag', function()
{
    printf( '><div class="form-field">%s</div', esc_html__( 'Some content', 'mydomain' ) );
}, PHP_INT_MAX );

1) Update:

I didn’t noticed at first that this is inside the <form> tag, so this is a hack to adjust to that and the PHP_INT_MAX priority is here to run it as late as possible. But I don’t like this approach, the {$taxonomy}_add_form_fields is more suitable, but then you would need to move it up via CSS (flex order?) or javascript.

It’s also possible to hijack the fourth esc_attr() call, after post_tag_term_new_form_tag is fired up, via the attribute_escape filter, i.e. the one inside the wp_nounce_field() but that sounds too unstable.