<p>I want to place a Webpart on a page that holds a subfolder of the Document Library in SharePoint, but somehow, the only thing I get is the root folder of the document library.</p>

<p>Is there a Webpart that fills this need?</p>

By default I don't think that is possible.
The list web part that would show the Shared Documents understands how to render the library, but doesn't understand how to filter to only show the contents of one subfolder.
It would be nice to create a Filter Web Part and to provide that filter to the List web part so that it filters according to the sub folder defined within the fileref field of the document library. However the filters it appears to be able to consume are Type, Modified and Modified By. So you could filter it to just the documents you touched, but not the ones in a given location.
End result: Roll your own web part.