The 00 globe goes back to the U.S. Safety Brass lantern made by Dietz and S.G.&L. starting in 1881. After the Dietz lantern tooling was destroyed by fire they took over the S.G.&L. plant in Syracuse in 1897 and made lanterns with no maker's mark, using the existing S.G.&L tooling. I believe this is when they started using Dietz marked globes. The U.S. Brass was phased-out 1902-1905, being replaced by the brass Junior cold blast in 1902 and the tin Little Star hot blast by 1905. I believe the early Juniors would have also used the 00 globe.
The correct globe depends on the date of the lantern. The 1913 catalog shows it with the 00 U.S. Tubular globe. The 1917 catalog shows the Little Star globe with the cold blast shape. (I don't recall ever seeing one of those) The 1926 catalog shows the Little Star with the Junior Loc-Nob that came out in 1923. If your lantern has the broad-bead vertical tube reinforcing, it is after 1915. If it has the spiral wire burner, it is 1920 or later. I'm looking forward to a picture!