No, adding more would be bad. What you want to do is to clean off all existing paste (use isopropyl alcohol if you can) and apply a bit of fresh paste.
If you're talking about the layer that comes with a new cooler, you can usually use it directly - you don't need to use your own at all. Replacing paste is only really worthwhile for old paste.
Also, the correct saying here is "less is more"1 :)
With thermal transfer from the integrated head spreader (IHS, the metal on top of the CPU die) to the heatsink, it vaguely goes:
- metal-to-metal contact: best
- metal-paste-metal contact: alright
- metal-air-metal contact: very poor
So your best-case scenario is if you can maximise direct metal contact between the IHS and the heatsink. That means they should be as clean and smooth as possible, and a fair amount of pressure pushing them together.
Now, if direct metal contact is best, why do we have paste? Because it's very difficult to get solid metal smooth enough for perfect contact, so you inevitably end up with lots of tiny air bubbles, resulting in poor transfer. Adding paste fills up these little gaps, but adding too much paste1 will either form a thick layer and prevent direct contact, or will end up getting squished out the side.
Even worse is trying to apply fresh paste on top of existing old/dried paste - that way you have the poor performance of dried paste (which can no longer spread effectively once disturbed) plus an additional layer. It's much better to just clean off the existing gunk first.
1 You'd want enough paste. There's a bit of leeway here, but you also don't want to go squeezing a whole tube in - once you have enough, adding more won't help. Keep in mind that what looks like a tiny bit will actually spread out quite far once pressure is applied - you're squeezing a 3mm-high blob into less than a tenth that height. Optimally, you'd have somewhere maybe a little bit over enough.
For those interested, there's further discussion of specific application techniques and their relative effectiveness here: https://www.pugetsystems.com/labs/articles/Thermal-Paste-Application-Techniques-170/
Best Answer
If you follow the above mentioned criteria;
You should be fine. However, I'd like to touch on a few things, beginning with the anti-static wrist bands and overwhelming amount of blame a static discharge gets. I've been working in the electronics industry for well over 15 years, in all my years, I might have come across 10 instances where static electricity could have been the cause. Am I saying you shouldn't invest in wrist bands? No. What I am saying is, its good to be aware of the potential threat a static discharge can have on motherboard components, but I wouldn't wrap yourself in static bags and anti-static boots, and clamp yourself to an anti static wrist band.
When he/she said "Use very little thermal paste" it's no joke. Believe it or not, if you coat the cpu chip just enough to where you can see the information on the chip through the thermal paste, you got it on just about right. Less than paper thin, almost a hazing, is good for most CPU thermal paste applications. Typically, if I'm going to clean and apply new thermal, I first unseat the CPU, clean it thoroughly with alcohol or other non-aggressive cleaners, get everything shiny and new (including around the edges). Then, I reseat the processor chip, apply my paste, and put it all back together for a test run.
The majority of the time its a success and your off to the races. Sometimes though, things go wrong, when that happens, remember a little less thermal is never a bad thing.