No, unfortunately it won't be quite that easy, but it is not hard to do.
Essentially, what you will want to do is:
1) Using the easy one-click installation WordPress admin tools, export all your content. Save this export file on your own computer.
2) If you want the new "advanced" installation to be in the same place (at the same URL) as the existing installation, make sure you have saved all your uploaded assets that you have uploaded into the existing WordPress (images, etc.), or have them available on your own computer for subsequent uploading to the "new" installation. Once this is done, use the web control panel to delete the existing easy one-click installation.
Of course, if you are installing the new advanced one-click WordPress installation into a new location (URL) and you plan to keep (at least for now) the existing "easy" one-click installation online, you can skip to step 3 if you want because you can get these assets from the existing installation later.
3) Use the web control panel to create the new "advanced one-click" Wordpress installation.
4) Import the database content that you saved in step 1 above to the new WordPress installation using the WordPress admin back-end tools.
5) Run through the settings in the WordPress back end of the new installation and duplicate what you had before, or change as desired.
6) Re-upload the assets you preserved in step 2 above into the new WordPress installation. This time you can use FTP to do that if you know where everything was - it is almost always going to end up in the "wp-content/uploads" subdirectory
7) Regenerate your permalinks.
... and that's about it!
--DreamHost Tech Support