I never bothered to ask - what is your primary use of this PC? Judging by your parts it's likely a casual gaming and web browsing PC, but I shouldn't make assumptions.
Regarding multithreaded performance, an i3 would not slaughter your current chip. Rather, it would probably lose, though not by much, but you knew that. My experience has been that two fast cores with HT make for a better system for those purposes, hence my recommendation. Having used both (I have multiple AMD and Intel systems in the house) I'd take an i3 over any FX for casual gaming and web browsing.
It's definitely a factor that you already have a cooler that would allow for a decent overclock, and probably a power supply to support it too. These are items that add cost to a from-scratch AM3 build and help to level the playing field, but having a cooler doesn't necessarily mean it needs to be used. I have boxes of old parts unused and I don't feel compelled to stick them in my current computers. An overclocked FX-6300 would slaughter your current system in multithreaded performance, but unless you're shooting for 4.5+, would probably be a regression in single-threaded performance, and thus you'd (marginally) lose FPS in some games.
Perhaps the deciding factor would be this - would you rather have a Phenom quad chip at 6Ghz, or one clocked slightly lower than your current one but with more cores? Which would benefit you more?