According to the Elgin 1958 service manual, all postwar Elgin 8/0 size movements take the same stem. By the time the 1958 manual came out, they stopped using the part numbers, and instead listed them by 'basic grade', so your 680 takes a '532' stem. Dashto has 'em.
I believe it's tap size 9.
BTW, those 1950 and 1958 service manuals are golden. You can usually pick them up on Ebay for about $10, and they're loaded with great info.