Adobe does not recommend having previous versions of both Acrobat and Reader on the same Win machine. And if you already deleted one, and now intend to reinstall it, the usual recommendation is to remove all traces of both before doing a clean install of the one you want.
Acrobat 6 is now 4 versions back too. Another problem you'll start having is with documents that aren't compatible that far back. If you download something and want to edit it, and 6 won't open it, you'll see what I'm talking about.
You can do what you have stated as well i.e. Uninstall Reader 9 and reinstall Acrobat 6 and then Reader 9. But do make sure that while installing Reader 9, select Reader 9 as the default PDF viewing application when you are prompted.
Or, once you have installed BOTH Acrobat 6 and Reader 9 on your system, please change the following registry key to point to AcroRd32.exe (usually present at: "C:\Program Files\Adobe\Reader 8.0\Reader\AcroRd32.exe").
Hope this resolves your problem