A larger lumber repeatable will open after the 8th quest of Maahes.
Some time after reaching Scandinavia, another lumber repeatable will open.
The focus on lumber to open slots in the underground camp will eventually switch to stone.
There isn't really a need for much lumber in China, but there is in Atlantis. That's as far as I have gotten.
You can send caravans for lumber, but it's an expensive way to get it while you still need the coins for equipment.
You can look for an older (100 gem) story and choose one based on the amount of lumber. As I recall, Jungle Book is a small event that is good for lumber. However, we have Halloween 2020 now, and will get Halloween 2019 in a week or two. It might not be the best time to buy another story.