Does the hole go all the way thru to the other end. I have used fishing sinkers shaped like your piece. They had a wire that went thru the weight with a loop on the top.
Was it found at a CW site? Perhaps it's a carved bullet. Bored soldiers loved to carve on lead. If so, that's one of my favorite finds. Look at it with a loop and see if you can see any knife marks.
Conical bullets invariably have a length of parallel section where they contact the rifling/bore ("bearing surface"). Your object is curved along the sides.
I’ve dug thousands of Civil War bullets and never one like this. The area I dug it in has Indian colonial and Civil War relics. Just an odd piece for sure, I thought it was an egg fishing weight myself But doesn’t appear to have a through hole but I suspect it did and the tip been pressed over closing it off at some point. That’s my best guess I appreciate the help.