The top item looks like a Gardener cleaning round. I can't tell by your picture if it's 58 or 54 caliber.
The bottom item is a worm for attaching to the ramrod of your musket. it can be used for removing a round from the barrel, or a patch that fell off from cleaning or even wet and damp packed in black powder. Nice find. And since it was found at Corinth it probably was Confederate, since this was a depot for supplies along with a muster area, and camp.