Buy a ball of twine, preferably some sort of
tough plastic twine, and tie the end very securely to one end of the existing cable. Now yank the cable out of the wall from the other end. Hopefully, you'll end up with a piece of twine running where the cable was.
Now tie some Cat5 cable and another piece of twine to the end of the piece that's running through your walls, and pull that back through. Now you have a new piece of cable running through your walls, along with a piece of twine that you can use to pull another cable in the future, should you decide you need something else.
I make it sound far easier than it is, but it's worth a shot.
Also, when you say you suspect that the current cable is Cat3, you do know that Cat3 is still twisted-pair, right? It's relatively uncommon to find that in anything but fairly new homes. Chances are that it's just plain flat 4-conductor telephone cable. Which means that it's rated at far less than Cat3, and doesn't have enough conductors anyway. (100BaseT4 requires 8 conductors.)