No, you don't need to upgrade the cooler. The stock cooler for the i5 doesn't naturally give those high temps; of which they are. Before you go spend more money, I'd try reseating the CPU, as well as double check to make sure the fan is 100% installed. Even a slight tilt for the fan will cause a non-full connection with the CPU and will result in high temps. Also, check to see if the fan is connected properly to the motherboard. You might not be getting full power to it, causing a slower fan speed.
I have an i5 in one of my rigs and before I bought my heatsink I used stock cooler and my temps were idle @ 45-50c and under full load around 60c. You shouldn't need to get an aftermarket heatsink, especially if you're not even OCing.