We already have four (technically five if you count chain) armor types, the problem is that only two of them are used.
Leather armor and gold armor need some changes IMO.
Firstly, gold armor could be made actually useful if it was given high protection. Keep its durability just as low as it is now, for balance (and because it's in the spirit of gold tools not being durable), but the protection could be put up to, let's say, 9 armor bars, so it's better than iron but still worse than diamond.
Secondly, leather armor is useless since iron is even easier to find than leather in most cases, all you have to do is dig down and/or find a cave or two and you can get the 24 iron required for a full set easily, whereas cows are scattered around the map and in too little numbers to get 24 leather swiftly, instead you would have to breed them, which takes way more time than getting iron.
Either we could make cows spawn more often, and possibly drop a little more leather, or we could add another source for gathering leather, for example, other mobs rather than just cows could drop it (which would be realistic since 15% of humans' leather supply comes from sheep and 11% from pigs), or it could be found occasionally in the ground, I'm not sure.
Either way, something should probably be done IMO, because while leather is fun to make as you can dye it pretty colors, it's next to useless, and gold armor needs a lot more than good enchantability (and being the prettiest thing in existence) to be worth it.