If you make the brain larger, some things will get worse, rather than better. The cost of communication will be higher, it will get harder to dissipate heat, and so on.
It's quite possible evolution already pushed our brain size to the limit of what actually produces a benefit, at least with the current design of our brains.
The more obvious improvement is just to use our brains more. It costs energy to think, and for most of human existence food was limited, so evolution naturally created a brain that tries to limit energy use, rather than running at maximum as much as possible.
Convenient for your hypothesis, some research claims that Neanderthals had brains larger than our present day average.