I've tested this a number of times to be sure.
When you first begin, take a look at the available guildleves and the listed item rewards. If you start new character, you will have similar leves available but different item rewards.
Personally, I am really digging the pointy hat. In Limsa, the Hempen Hat is a possible a Rank 1 Battle Leve reward, but not always. I've actually taken to restarting my characters to make sure that the hat is listed as one of the available rewards.
Just pointing this out to others that may want the pointy mage hat early on as well, and also to clarify that the rewards you see will not necessarily be the rewards that your friend will see for completing the same leve.
In FFXI you had to wait for 52+ levels to get a pointy hat! In FFXIV, you can get one in the first 30 minutes.