You get faction standing increase from storyline missions only. As far as I know, the security status of the system doesn’t matter. You will get corporate standing from both regular and storyline missions.
You can speedup the process by training social and connections skills.
For the epic arc you need standing with the faction or the initial agent.
i.e. the Gallente Arc “Syndication” needs standing of 5.0 with Impetus corporation or Gallente Federation before Roineron Aviviere will talk to you.