We've all seen it countless times. New world, a couple of muppets over hunting and bang, half the research cuts off. How about adding paddocks?
Full functionality proposed:
New skill: Herding
Herding Recipe 1: Herding Lasso from Fibers.
Herding Recipe 2: Paddocks (Varying sizes).
Paddock Functionality: Functionally similar to storage with smaller capacity but can only hold one type of livestock. Periodically if there are 2+ livestock present and a free storage capacity, it might breed.
Optional Extra: May require feed to enable breeding. Either fibers from grass etc, or add Animal Feed Recipe(s) with efficiency skills.
Lasso Functionality: Can't be used while carrying. Target similar to bow, but slow descent. On hit (% chance for animal to escape could be another skill saving), animal captured. Carried item becomes "led animal". This can be deposited in paddock or placed on ground at which stage it becomes a wild animal again.
A byproduct of paddocks might be animal waste fertiliser providing a small boost to fertility across the board in farm fields.