I think you really have to leave the FX platform in order to achieve better pairing with your GPU (as well as better upgradability). I don't see a point getting a CPU that fits your current build as the expenses for a new CPU in that platform is not worth it in terms of price/performance. Of course, migrating into a different (newer) platform means changing your CPU and your MB and your RAM.
If you want a cheap AMD build with good upgradability as well as overclockable, you'd be looking at these costs:
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 5 1400 3.2GHz Quad-Core Processor ($157.49 @ SuperBiiz)
Motherboard: ASRock - AB350M Pro4 Micro ATX AM4 Motherboard ($77.98 @ Newegg)
Memory: Corsair - Vengeance LPX 8GB (2 x 4GB) DDR4-2666 Memory ($76.99 @ Newegg)
Total: $312.46
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-11 02:21 EDT-0400
If you want an alternative affordable build, you can also go for an Intel non-OC'able build, such as this:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Pentium G4560 3.5GHz Dual-Core Processor ($78.89 @ B&H)
Motherboard: MSI - Z270-A PRO ATX LGA1151 Motherboard ($109.89 @ OutletPC)
Memory: GeIL - EVO POTENZA 16GB (2 x 8GB) DDR4-3000 Memory ($104.99 @ Newegg)
Total: $293.77
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-11 02:26 EDT-0400
The Pentium G4560 (non-OCable CPU) in the above build is a place-holder (which is paired with a OC'able Z270 motherboard and OC'd RAM). The point is to upgrade the Pentium G4560 in the future to an i7-7700K once you have enough budget for that.
Considering the two CPU's mentioned above (the AMD Ryzen 5 1400 and the Pentium G4560), both of those CPUs will be able to handle your GTX 1060-6GB GPU and both are higher in performance compared to your current FX 6300:
Ryzen 5 1400 is ~40% faster in effective speed than the FX 6300:
http://cpu.userbenchmark.com/Compare/AMD-Ryzen-5-1400-vs-AMD-FX-6300/3922vs1555
Pentium G4560 is ~25% faster in effective speed than the FX 6300: http://cpu.userbenchmark.com/Compare/Intel-Pentium-G4560-vs-AMD-FX-6300/3892vs1555