Put a clean sheet in the bottom of the hamper. put gifts in it and cover with another clean sheet.
Put in the trunk of the car.
small pkgs can be hidden inside laundry detergent or dryer sheet boxes, or behind cleaning products that nobody but you uses.
I opened the bottom of a cereal box and put a gift there once. Closed it with Elmer's glue and nobody noticed.
If you have a storage unit, that works.
Neighbor's house. But that can become complicated if the neighbor's kids think it's for them.
Have someone without children act as the neighborhood stash house for gifts. They would probably get a real kick out of it.
Long skinny things go behind the refrigerator or dryer.
tie the hem of hanging pants and stash things there. Or hang down inside of a formal dress that you haven't worn in years.
If your kids never clean their rooms, hid their gifts under their beds.
That's all I can think of.