Can display plugin without shortcode?

You would need to use a hook, but there’s no standard hook after the header, so one would need to be provided in the theme. But then your plugin would only support that particular theme. An alternative would to do use JavaScript, but you would need to know the markup of the theme, which also isn’t standard.

The best you can likely do is include a function in your plugin and provide instructions to site owners telling them to add it to their theme.