Actually, the 1060 does pretty well at most games, regardless of quality settings, at 1080p. Certainly there are a FEW games that break the mold, but overall the 1060 is very capable at 1080p. For 1440p or 4k, not so much. You'll want at least a 1070 for 1440p and a 1070 ti would be better. For 4k I think a 1080 ti should be a foregone conclusion if you want anything over medium settings.
I think the bigger question, that hasn't been addressed here, is what works best with that CPU. I think gaming performance will be hampered, at least to some degree, with any card tiered higher than the GTX 1060 with that CPU. I hate to throw around the term bottleneck because it is used so loosely these days, but there it is. That 4590 probably becomes a bottleneck once you move past the GTX 1060 UNLESS you are gaming at higher than 1080p and using very high or ultra settings in which case the CPU is probably going to get more time to do it's part in rendering the frames and likely won't be holding back the GPU from doing it's job.
Another consideration will be the refresh rate of the monitor. There is very good information to be found, if you are able to adequately parse the information, here:
https://www.anandtech.com/show/2803/3