This is actually really easy to do. It's 4 easy steps:
1 - Create a new directory in your root directory with the name you want to use
public_html/home
2 - Move all your Joomla files into it
3 - Open configuration.php and change the following line like this:
Find:var $log_path = '/home/user/public_html/logs';
var $tmp_path = '/home/user/public_html/tmp';
Replace with:var $log_path = '/home/user/public_html/home/logs';
var $tmp_path = '/home/user/public_html/home/tmp';
note: the first /home/user/ is related to the root of the server and most shared servers have accounts set up like this. All you really need to worry about is putting your directory name after the public_html/ part.
4 - Upload your new configuration.php file again and change the permissions back to 444
Now just visit your site through yoursite.com/home
Since you're staying on the same server you don't even have to worry about your database, just leave it as it is and the site will pick it back up.
Since this is a live site, you can copy the files over to the new directory and actually have both installations running at the same time. You don't want to do this for very long at all and you should have your other "splash page" files ready to put into the root directory once you verify the new site is up and running. Be sure to delete the first Joomla site files that you aren't using anymore once you're all set up.