This is just an assumption, but have you completed your Grand Company Hunting Log? This is something that a lot of players overlook or forget about, but it's a necessary requirement for getting later promotions - players mistake it for their class Hunting Log, but it's a separate Log (in the Hunting Log menu, it's accessed by clicking on the logo of your Grand Company, in your case, the Maelstrom).

There are only three tiers in the Grand Company Hunting Log, and they only reward Company Seals instead of gil and exp, plus some of the enemies you need to defeat are only found in specific dungeons, meaning you will have to run those in order to complete them. But it's mandatory if you want to attain the highest currently available ranks in your Grand Company (currently the highest attainable rank is Second Lieutenant, as Najara mentioned).