I think you can do that with a Page List block.

You can configure it to show the first level of pages underneath another page. If you point it to the home page, it should only display your top-level pages.

The design will look a bit different than the Auto-Nav block, which I believe is the block that's used for the default site navigation. You can either change the site's CSS or you can create a custom template.

If you're using "Header Menu" template for your navigation, the first li-element has class "first". If you're on the home page, it has ALSO class "nav-selected". Therefore it should be possible to hide the link with CSS.

#page #header #headerNav ul.nav-header li.first.nav-selected {display:none}

This may not be the best practice, but if you don't need that "first" class to something else, it shouldn't be that bad solution.

I'd concider twice if the home page link is really unnecessary there. It's good to show users where they are. In some cases it is redundant, but not always.