He also doesn't want to spend more than $250. And the GTX 1060 6GB can do 1080p60 very well. If it couldn't hold 60fps on The Witcher 3 with mods on Ultra, optimizing it slightly (such as lowering Shadows) should be worth not spending an additional $200 on a 1070.
Here's an idea for you, OP. Since your case can't accommodate a high-end graphics card, you could buy a similar-sized, console-like gaming PC case that's designed to accommodate one. I've looked into it before, and nothing has really interested me
yet (either too expensive, or not compact enough), but there might be something that suits you.
Example:
NFC S4 Mini (This one is small enough that a standard 1070 wouldn't fit, PSU is external)
Just an idea.