Console scripts, mods, or voodoo is acceptable as an answer.
First, back up your save games and then you can try using AutoHotKey to create quicksave backups. The way I've set up AHK, it should refuse to overwrite files, but I wouldn't trust your only copy of your Skyrim saves to this, as it's untested. Again, I provide no warranty, use at your own risk.
Try the following, using AutoHotKey:
Edit the paths as needed for the location of the quicksave file and where you want the backups to go. If you put the "BackupFileName" location as the save directory, then it will probably show up in your list of saved games, and they'll be named with the date/time you quicksaved previously.
Once you've edited it to your liking, you'll have to put this in a file with the extension ".ahk" and double click.
I don't have Skyrim, so I can't test this out personally. Hopefully it at least points you in the right direction.
The quicksave functionality in the game engine really isn't suited to making multiple saves, for a number of reasons having to do with loading optimisations and bugs related to that. Trying to make multiple actual quicksaves is a recipe for savegame corruption and crashes.
What you want is a "full" save, which makes a new save each time automatically, with the one-button ease of a quicksave. All you need is a way to trigger a full save with a button, and you've got a effective "quick save" button that cuts out the one-save-only and bugs of the game's built-in quick save.
There are a number of ways to do this. The easiest is to install a mod that uses SKSE to trigger the save command, such as Save Hot Key ArchMod. Any of a number of mods should do though, so that's just one to get you started. If you want to shop around, search the Skyrim Nexus for