[RedDevil]

Raising the tire pressure will influence handling in several ways.

Because of the smaller contact patch the angle of attack between the thread and the road will be less. this will cause less wear and less noise. But the sides will vibrate at a higher frequency, and those vibrations may therefore become audible.
Steering will be lighter, and so will stresses from non-neutral alignment. Your bearings may last longer.

The sidewalls will flex less; in corners the thread will have less sideways travel, increasing stability and grip. I can corner faster by running my tires 7 PSI over the door sticker pressure; there is one corner when I exit the highway where I want to conserve as much momentum as possible, without making the tires squeal. It may not make a big difference, maybe just 5%; but it is there.

The Gen2 Insight is said to be sensitive to sidewind. And in my first months of ownership I experienced that too - a distinctive buffeting rear shake.
It disappeared when I raised the pressure.
Then I had my wintertires mounted by my dealership. On the way home; there it was again. Pressure up, gone for the winter.
Spring came, dealership put the summertires back on and carefully lowered their pressure to just below the sticker pressure... you guess what happened next.
That was the last time.
This known phenomenon disappears if you run the pressure over the posted too-low door sticker value!

When I started reading posts on ecomodder I was sceptical about raising tire pressure, but I gave it a try if only to stave my scepticism with experience. It all went horribly wrong so now I switched to the other side