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Higher threat levels generally mean more abundant ores, and ores of higher rarity. However, especially rare ores like gold or diamond can still be hard to come by, generally occurring only at the deepest levels of the planet. Personally, I'm mining level 19 planets for gold and maybe producing 4 ores by the time I reach the planet's core. I found two gems (8 ores) worth of diamond in the 1-9 planets, and haven't seen any since.

That said, planet type may have something to do with it. I've noticed that moons contain a more ores than I'm used to, in addition to being a bit easier to dig through. Desert and arid planets are super easy sources of iron, copper and coal since you can just dig into the sand to dislodge everything.

Some advice: hoard your iron. You'll want it for Sector Beta.

There's a set of craftable clothing that you can make at the Spinning Wheel. Snow Infantry Gear. The flavor text suggests that it protects against cold weather.

The problem being it requires leather to craft, a total of 38 pieces for the whole set. Leather is somewhat hard to come by due to a low drop rate. I suggest heading out and hunting down animals with an iron hunting bow until you get enough leather for that gear.