Upgrading from Pentium D to C2D - which fan?

I currently have a Pentium D 940 with what I believe is the stock cooler for it. Looking at the specs for my motherboard, I saw that I could put certain Core 2 Duo models in it, one of the fastest being the E6600, which would be a huge boost in performance for me.

From what I've read, Pentium Ds put off a lot more heat than then the Core 2s, despite the Core 2's superior performance is that right? With that reasoning, would I then be better off using my current cooler for the E6600, as opposed to the stock cooler that comes with it? I am planning on overclocking it some, but not that much. I'd rather not spend another $50 on a cooler when the one I have works just fine.

Any feedback is highly appreciated, especially if I'm overlooking something major here. Thanks.