What? You need Second Lieutenant, not First. First Lieutenant and Captain are completely optional. The only things you get from going further is being able to buy Captain Weapons, the Lieutenant Hat/Coat, the company's 2 Orchestrion Rolls, Company Aetheryte Tickets, and the Materiel Containers that have a 0.001% chance to have a Night Pegasus Whistle in them.
To do this, you need to complete your Rank 1 and Rank 2 Company Hunting Logs, as well as Shadows Uncast (Dzemael Darkhold) and Gilding the Bilious (The Aurum Vale).
While it's annoying to get all of this done quickly as a new player or someone who bought a story skip, it takes less than 2 hours to farm up the seals needed to buy yourself through the ranks and unsync the 2 dungeons.

Just start "Let the Hunt Begin" at your Grand Company's city, and boom. Now you can do the HW hunt unlock chain, and then unlock the Stormblood, Shadowbringers, Endwalker, and Dawntrail hunt boards.