The problem you have here could be due to noise coming on to the cable from outside sources, or from noise coming onto the cable from the "tape" or whatever player you have on the end of the 3.5mm jack. If the noise is from an external source, you will want to shield the cable, if the noise is from the devices at either end of the cable, you will want to put a clip on ferrite on the cable.
Grasping it with your hand has two effects on the signal up the wire: it creates a shield around the wire and it puts a ferrite clamp around the wire. This because your hand is thick enough to stop external signals, as well as mainly water, and so can act as a ferrite.
As the foil had no effect, I would expect it to be from inside one or other device. Often I would blame it on trying to over amplify the signal from your source, but as grasping it removed the noise, I'm more of a mind that it's just some EMI leaking out.
My first action would be to get a clip on ferrite that I happen to have getting in the way in my tool box, and put it on. But that isn't a quick solution, unless you happen to have some. Farnell/onecall/RS/Mouser all stock something suitable, such as this page from Farnell , but I assume you're in the USA, and I don't know what the high-street electronic stores are like over there. (Minor detail to note: you may need to be careful as to which one you get; you want to block the noise (at whatever frequency that happens to be) while making sure your desired sounds don't get blocked.)
As for wrapping the wire multiple times in the clip on ferrite, doubling it has an effect, but after a third pass, there's not much benefit (though WURTH Electronic have some interesting graphs of the performance). There is a bit of debate about where the ferrite should be put; source end, middle or receiver end. I would say receiver end would have most benefit for your problem (as you're trying to stop noise getting into the tape), but if the noise is coming from the tape device, this could make it worse (the noise may get reflected back up the cable into the player).
In short: get a clip on ferrite, add it to the end nearest the player.