The internet can be a tremendous resource if you know where to look. But can take a lot of practice for your searches to turn up what you want. One trick you can use is to run more searches for specific details. For example, you've probably searched for "Mapp vs. Ohio" and gotten a lot of stuff you already knew. So try running a search for the exact details you want.
Another thing you can do is to run more general searches than exactly what you're looking for. I tried a search on Google for "supreme court" and archive. This gave me some links to sites with archives of past supreme court cases. Most archive sites are searchable, which is where I found this link that may be useful for you.
However, you should be aware that the information you want isn't always going to be out there. When I was doing my search, I found an official archive database by the government including all the details on cases going back to 1992. But cases before that aren't there because the internet was not really established and actively used until around then. Since the case you're looking for occured in 1961, you probably won't be able to find a 'play-by-play' description online. If you need that much detail, you'd need to contact the federal court archives directly.
(published on 10/22/2007)