How do I get more rare crafting ingredients?

Take a look at the Apprentice Leather Doublet

Underneath the stats, you will see Salvages Into section. This will show you your chances of retrieving crafting supplies. For this item, you have a 100% chance of getting a Subtle Essence, and a 15% chance of a Fallen Tooth. This information is not shown in-game; as far as I can tell, only the website has these references.

Unfortunately, I couldn't find stats on dropped magical or rare items, so I have no concrete proof that they conform to these standards.
However, crafted items work as follows (and I believe dropped gear would work similarly):

• Magical: 100% chance of "magical" (blue color) salvage, 15% chance of "rare" (yellow) salvage.
• Rare: 100% chance of both magical and rare salvage.
• Set: 100% chance of 3 rare salvage.
• Unique/Legendaries:
  • Below Level 60: Treated as Set items, so 3 rare salvage.
  • At level 60: See below

I was poking around the blacksmith's recipe's, and realized: Inferno has an extra salvage! Using the same source, Inferno class gear results in the following:

• Magical: 100% chance of "magical" (blue color) salvage, 15% chance of "rare" (yellow) salvage, and 0% of Legendary salvage.
• Rare: 100% chance of both magical and rare salvage, and 0.1% of legendary salvage.
• Set/Unique/Legendaries: 100% chance of 2 rare salvage, and 1 legendary salvage.

UPDATE: As of 1.0.3, Blizzard reduced the chance of salvaging Fiery Brimstone from magic items from 0.8% to 0%.

If you're looking for non-Inferno rare salvage (Fallen Tooth, Lizard Eye, Encrusted Hoof), your best bet is to either:

• Salvage rare gear found in the appropriate difficulty.
• Buy the material you are looking for from the auction house. This is the probably the cheapest option for Normal and Nightmare currently.

Since set gear doesn't drop before Inferno difficulty, and any crafting of said set items will take more materials inputted than received, this isn't very viable. You can salvage any uniques you find, though, if you don't want them anymore. You'll get more return selling it on the auction house and using that cash to buy what you're looking for, though.

Now, the tiers of materials. The general idea is that each difficulty has it's own salvage materials, magic and rare. In practice, it's not quite so clear cut. I've salvaged items found in Nightmare and received Normal difficulty materials. Testing has revealed the following results:

Armor/Weapons:

• Normal: 1-29
• Nightmare: 29-49
• Hell: 50-59 (Exclude max level relics)
• Inferno: 60+

Jewelry:

• Normal: 1-26
• Nightmare: 27-47
• Hell: 48-58
• Inferno: 59+ (Max level relics are included)

Source: My testing

So, when salvaging an item, make sure to see what the required level is. If the item has a reduced requirements affix, then add that to the required level to get the true item level. Since 1.05, and the introduction of Monster Power, Reduced Level Requirements affects the materials gained. So for inferno salvage, your items cannot have a reduced level requirements affix.

http://us.battle.net/d3/en/item/fallen-tooth

You should be disassembling rare items for rare ingredients (normal rare for Fallen Tooth). I read a link earlier about having a smaller chance to get them from magic items, but can't find it right now.