I don't think anyone knows how or if these things work. I finished all ranks for both the Feds and the Empire and I can't tell you anything of great insight. There doesn't seem to be a discernible pattern or anything consistent about how/when the missions pop up and whether or not doing one directly means you're going to move up after completion. For me, sometimes it did, and sometimes it didn't.
I'm Allied with both the Empire and the Federation and I have spent the last two days doing missions for minor factions in Alliance territory trying to move from Friendly to Allied. I don't see any indication of that happening soon. I'm Allied with a few minor factions, but still only Friendly with the Alliance major faction.
I'd love to know if anyone has ever achieved Allied status with the Alliance...
The whole Rank and reputation system seems to be just one big mystery that few understand. Doing missions for minor factions allied to the major faction is the only clue I know to getting the needle to move. It can take a lot of time to pass doing missions to get anything to happen.
Good Luck