For my particular particular Carolina rig set up, a 5/8 tungsten weight (I prefer the barrel swivel type), would be ideal. I can’t seem to find that anywhere. Does anyone know of a source?

Omfishingsinkers.com

Woo! Tungsten sells a 5/8oz flipping weight. I'm not sure of anyone who makes a 5/8oz barrel weight. Because of the nature of tungsten and the fact that it's made overseas still, the market isn't huge for it. 

Looks like 1/4+3/8 is your option. You could hold them together with shrink tubing.

Tom

The also have a 1 oz

Pretty much everyone sells the standard sizes. I have some 5/8 barrel weights in lead. Seems like that is a sweet spot weight wise and someone would make that size...

Not to high jack this thread but 5/8 oz is also a good jig weight that isn't popular.

Tom

I know. I have some. I don’t understand why more people don’t make that size. They make a 5/16. Seems like an in between size would be doable.