If your objective is higher fps, then you need to find your current limiting factor.
Some games are graphics limited like fast action shooters.
Others are cpu core speed limited like strategy, sims, and mmo.
Multiplayer tends to like many threads.
You need to find out which.
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To help clarify your CPU/GPU options, run these two tests:
a) Run YOUR games, but lower your resolution and eye candy.
If your FPS increases, it indicates that your cpu is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely more cpu limited.
b) Limit your cpu, either by reducing the OC, or, in windows power management, limit the maximum cpu% to something like 70%.
Go to control panel/power options/change plan settings/change advanced power settings/processor power management/maximum processor state/
This will simulate what a lack of cpu power will do.
Conversely what a 30% improvement in core speed might do.
You should also experiment with removing one or more cores/threads. You can do this in the windows msconfig boot advanced options option.
You will need to reboot for the change to take effect. Set the number of threads to less than you have.
This will tell you how sensitive your games are to the benefits of many threads.
If you see little difference, your game does not need all the threads you have.
It is possible that both tests are positive, indicating that you have a well balanced system,
and both cpu and gpu need to be upgraded to get better gaming FPS.
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If you need faster core speed, the i7-7700K is a decent option, even if you can not overclock on your motherboard. A used 7700K runs about $200-$250 on ebay.
If you simply need more threads, then either 7700 or 7700K will be a decent upgrade.
If you need more than what a 7700/k can offer, then you are looking at a platform change that will entail not only a processor change to ryzen 3000 or intel 9th gen but also a motherboard and ddr4 change.
I think with a 7700/k upgrade, a better than stock cooler would be a good thing.
A simple tower type cooler with a 120mm fan should suffice.
If you have 160mm available in your case, there are many very good options.
What Is the make/model of your psu?
That is what will determine your graphics card upgrade options.
Your RX580 is a power hungry card.
If your psu can handle it, it should also be able to handle a graphics upgrade like a RTX2070 or RX5700