Stock Intel i5 Cooler

I'm building an i5 system for a course of mine, and this is my first time actually building a system (though I have lots of experience inside PCs). The other day I fired it up for the first time, and while I was messing around in the BIOS I found the page that displays the hardware temperatures and it said the CPU core temp was a whopping 98C. It was only running for a minute or two. Immediately I shut it down and opened up the case, to find that one of the pins on the stock Intel cooler was a little loose. Checking the back of the mobo, the two pins on the bottom were in fine but the two top ones were not. I was able to tighten the two top pins without having to reseat the cooler, but the black middle pins aren't as deep as the bottom ones. I restarted the computer and running for 10 to 15 minutes the BIOS reported the temperature to be around 55C- better, but still seems a little hot to me. I have an i5 2300.

TL;DR: Messed up installing the stock Intel cooler, fixed it and it lowered the temps to 55C. Should I take the cooler off completely and re-mount it? Would I need to put new thermal paste on? The computer itself has only been running for 20 minutes at most and that was in the BIOS only. No OS installed yet.