I recently learned about Supernatural Hit Die over on the purple server. It's a concept that starts over at the
Necropraxis blogand got picked up by Luke Gearing for his Monsters& book. The basic gist is that monsters with SHD (supernatural hit die) don't really have hit points; each of their HD counts as 1 'hit', with the monster dying when they've run out of hits. The catch is that for a hit to count, it needs to reach a damage threshold. Otherwise, the monster simply takes no damage. For games with d6 weapon damage, 6+ seems to be a good threshold.

What this basically does is create the feeling that some monsters are just on another level entirely. It really does fit the original goal Necropraxis had for them, to simulate the sort of dynamics you see in Berserk.

Zombies are traditionally a fodder enemy. At low levels of OSR-esque DND, they are a bit scary, but no different than any other 2 HD creature. It's strange when you consider that zombies are typically considered very, very hard to kill. Shoot the head, destroy the brain, that sort of thing. Supernatural hit die model that very well, I think.

If you give zombies 1 SHD, they become much scarier. At low levels, all but a fighter with a big fucking sword is going to want to face them directly. They're still slow and stupid, but they become much more of an obstacle for the party to deal with. Encounters become reminiscent of the early parts in Resident Evil games, before you get the shotgun or the grenade launcher.

I think you could also work in a targeting system for them, though it does add a bit of complexity to combat. Players should be able to target limbs, with a damage threshold of 4+. It's still risky, but more manageable. Each limb should give them some sort of advantage. A leg might make the zombie slower, an arm might drop their damage. A legless zombie probably becomes much easier to kill, needing a hit but without a damage threshold. You could also ignore all of this and just reduce the damage threshold for each limb taken, or just mix both approaches together.

Other situations require adjudication.

What happens when the players use fire? I'd probably either lower the damage threshold for further attacks.

What happens when the wizards use magic? Honestly, direct magical attacks should just kill. Magic is incredibly powerful, and they're using resources to deal with your 1HD enemy.

If there are plenty of ways for the players to use their creativity, I think zombies become a pretty fun experience for everyone involved.