To upgrade your camp in New World, you will need to reach at least level 15 and start the quest Friends and Fashion in Monarch's Bluff by speaking to Bercina Thornton. Bercina's quest will require you to gather tattered silk from skeletons. You can get tattered silk from skeletons at the location antares and achernar.
You can also just follow the quest marker, and it will take you towards the skeletons. Kill and gather the silk from the skeletons and then take it back to Bercina to turn the quest in.
After you turn the quest in, you will then need to head to the town of Everfall and speak to a man called Odnell Lee. Doing so will complete the final part of the quest called Survivalist of Everfall and reward you with the level two upgrade to your camp.
Once your camp is upgraded to level two, you will be able to craft more items, and the cost to make the camps will also stay the same as level one, requiring both greenwood or flint. However, if you don't have either greenwood or flint, you can get both resources with ease by finding them out in the wild; it also may be worth keeping some on hand just to make building camps easier.
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