Tier 1 GC hunt log + the pvp intro mission (you only need to talk to an NPC) will drop a bunch on you. Sadly not enough to get you to Sgt. You could try farming dungeons as an Adventurer In Need (effectively only tank or heals). If that's not possible your next best bet is to do GC leves until Sgt 3rd Class. The amount dropped by ARR FATEs really aren't worth the bother.

Expert deliveries don't unlock until Sgt. 2nd class but you now have access to the Tier 2 hunt log. That should be just enough to get you to S2C.