Yes, you can put more on the page and there are a couple of ways to do that.
First, there is a status shortcode that can display content based on login status (in or out).
[wp-members status="in"]Your content here.[/wp-members]
[wp-members status="out"]Your content here.[/wp-members]
The caveat on this particular page is that at this time, these can only be used above the page shortcode. Additionally, this will show on the page when the links are shown, but also when you are editing the profile. But, it's easy to implement.
The other alternative is that there is a filter hook for the link content: wpmem_member_links. You can apply this filter with add_filter in your functions.php file.
That will give you the most flexibility, although if you dump the original links in the filter (such as you don't want them in bullets), you must make sure you still provide the links in some way so users can edit their profile and change their password.
Here's an example that keeps the original links and puts some text in front and behind:
add_filter( 'wpmem_member_links', 'my_member_links' );
function my_member_links( $string )
// this example retains the original links ( $string )
// and puts some text before and after
$string = "<p>I put this before the original links</p>"
. "<p>This is extra stuff I added with a filter</p>";
There's some more info here on this and some similar hooks, and there is a full list of action and filter hooks in the users guide available here.