That’s an excellent question. The simplest solution would be to use a plugin like Recent Post Extended to add in a customized list of your latest posts to one of the three widget areas at the bottom of the home page.
Another option would be to do a bit of editing and add in a list of those posts below the widget area. Here’s how to do that if you’d rather go that route:
Go to the Appearance section in your WordPress dashboard and choose Edit.
From the list of files on your right, choose page-home.php and find this line near the bottom:
</div><!-- #home -->
Right above that, add in this:
(Copy that code directly from the support site instead of the notification email.)
This line at the bottom of that new code block will add a button that you can use to direct users to the rest of your blog:
What I would do is go to the second page of your blog posts and copy the URL to that second page. Then replace the http://yoursite.com/blog with the URL to your second blog page. That way users can click that button and skip over the posts that they have already seen and go directly to ones they haven’t.
Save your changes.
I believe that will give you a list of posts below the widget area there. If you run into any questions or trouble, let me know.