Note that there are different standards for hard hats.

As noted, the lightweight plastic shell type (ANSI Type I) doesn't do a lot. It's not much more than a bump cap, though that's still better than a ball cap.

The CSA-approved Type 2 hard hat, for example, has substantial side impact (lateral) protection all around. These are required on all industrial and construction sites up here. They are heavier and more expensive, but designed to protect against concussion, so well worth it.

I believe there's an ANSI Type II as well, though the testing methodology and underlying assumptions are quite different.

EDIT: It's also worth considering whether your respiratory protection system will work with your head protection. Industrial hard hats and respirators are commonly designed to work together.