First, define the use for the hard hat.
For example, are you outside, in the weather ... or indoors? Are you typically in somewhat cramped quarters?
Outside, in the sun, all day ..... some amnner of full-brim, even one of the 'cowboy' hats, are the way to go.
Ordinary construction duties? Look for a hat with a 'longer' brim, and a rain gutter that directs water away from your neck and out, over the brim.
Inside, in an industrial setting? Look for one with a short brim.
Hot days? Look for a vented model; those little holes make a big difference.
The suspension matters- a lot. Six points are more comfortable than four. Ratchet adjust is much better than snap-ins. A foam brow band is better than smooth vinyl.
If you're in circumstances where the thing just keeps falling off, you might want to look at other types of protection, such as rock climbing helmets. That's another discussion.
Other attachments? Some hats allow you to clip on accessories, like face shields and ear muffs. Others don't.
Probably the biggest factor is weight ... but even the heaviest will stop bothering you once your neck muscles build up.