First up, your rep percentage with the Federation (Allied, Friendly etc) is largely irrelevant to rank, not even sure what that rep does, if anything.
Secondly, in order to move up the ranks (Cadet etc), once you're at 100%, minor factions aligned with the Federation can start offering missions that will award you the next rank in the navy on completion. These missions will always have words such as "Federal Navy" in their title. If you're still not sure about one, look through the rewards; if it's an ascension mission, it will have a rank at the bottom of the rewards.
It's known that you need a minimum rep level with a minor faction in order for ascension missions to spawn, however I'm not sure if its Friendly or Cordial, sorry (someone may be able to answer definitively). Once you finish the mission, you'll get the next rank.
Although FDev have buffed the spawn rates considerably, it's all RNG at the end of the day. Sometimes it may take a few boards to get the mission to spawn, and sometimes it will spawn a mission you're not confident performing. No need to worry about lost rank however, even though your panel says "100%", your rank isn't capped. When you do finally get the ascension mission, any rank gained since hitting 100% will be applied to your new rank.