When your Pig Pen can't contain anymore pigs, your piglet will be temporarily stored in "Waiting List" in the bottom menu of the Pig Plaza.
They'll automatically be moved to the Pig Pen once there is room; for example, when a pig has been sold. Piglets that were waiting first are moved to the Pig Pen first.
Only three piglets can be stored here. If there is a fourth waiting piglet, you will need to pick one to part ways with.