An integrated circuit package may have a package substrate with a surface to which an integrated circuit die is soldered. A first set of contacts on the package substrate may mate with contacts on the integrated circuit die. Solder may be used to connect the integrated circuit die to the first set of contacts. A covering material such as a plastic mold cap may be used to cover the integrated circuit die and the first set of contacts. The mold cap may have a rectangular shape or other footprint. A rectangular ring-shaped border region or a border region of other shapes may surround the mold cap and may be free of mold cap material. A second set of contacts on the package substrate may be formed on the surface in the border region.