How tall are the cones you need to store?
For embroidery thread storage, I had dry erase board cut to fit in the drawers. Then glued golf tees in a grid to fit the cones. It keeps the threads from sliding around when the drawer is opened. The tees come in different heights so this method could also be used for the large cones.
Each tray is stored in a wire basket drawer system and holds 81 cones.
I have also used trays like these in the double height storage boxes for serger thread cones.
https://www.artbin.com/cone-tread-tray-6901ab.aspx