I discovered that two strands of trimmer line, round, size .065", also works well for the last crocheted round of the brim. I have had good luck with the lamp wire when I separate the two pieces, and the trimmer line. Both are stiff enough to hold the shape of the brim.
Now, I have to find a better way of connecting the ends of both, besides taping them. Any ideas? Some kind of clamps, like in jewelry making, maybe? Or???