It depends on the games you play and at what graphics settings you use.
Generally, most AAA games (graphic-intensive ones), will only output less than 60fps on 1080p with everything maxed out (all the eye-candy bling-blings). Decreasing in-game graphic details will, of course, increase your fps output to ~60fps (depending on which games).
On other titles that doesn't use much GPU power (such as CS:GO, LoL, DOTA 2, and the likes), the GTX 1060 would be more than enough to reach 140+ fps on your monitor, with maxed out details.
Ideally, I recommend for 1080p/144Hz the GTX 1080/Ti (for AAA ultra, think: Witcher 3 or GTA V or Fallout 4 maxed out) or the GTX 1070 (if price/budget is limited). However, in your case, the i5-3350P *may* hinder the GTX 1080/Ti's max. possible performance (https://docs.google.com/spreadsheets/d/17VRKPjyiTBx9Ewc2xkmaMZD2tA3gSOG3rNtH4OEiz3g/edit#gid=0).
The GTX 1070 is your next and only choice (after the GTX 1060) to go with your current CPU (https://docs.google.com/spreadsheets/d/1EQOWVLxk0DOFXKfzCmz9lEZ-qpMb7mIvt8KUYek69A8/edit#gid=0) without "bottlenecking", if your budget permits for such powerful GPU.
Else, the GTX 1060-6GB will serve you well, especially coming from a GTX 660 (a very significant upgrade in itself, http://gpu.userbenchmark.com/Compare/Nvidia-GTX-660-vs-Nvidia-GTX-1060-6GB/2162vs3639).