Upgrading a 2012 gaming PC

Hi,

I've got a CyberPower PC from around 2011/12, and it's starting to show its age now. Although I can run things like BF3/4 on high settings at 1920*1200, things like No Man's Sky have to be run in a window at 1680*1050 to get smooth/playable frame rates.

It's still technically a decent PC, but I've been out of the hardware loop for quite some time now and what I need to know is if I can just upgrade the GFX to get that boost I need, or would the rest of the rig (4 year old chipset/slower bus speeds etc.) bottleneck the available speed?

Rig is as follows:

i7 2700K (OC to 4GZ on air)
8GB DDR3 1600
P8Z68-V-Pro
GTX 580
Asus Zonar
Cooler Master 850W Silent Pro PSU
Windows 10 64bit

I've not got a huge budget on upgrades at this time, so thinking about getting either a 970 or a 6gb 1060. Would this be a worthwhile upgrade to hold me over until next year do you think?

Any guidance would be appreciated. Thanks for looking.