It's the Military Academy that gives +1 SOC on Honors, not regular University, just FYI.
Entertainer and Scholar are the only careers that give +SOC as a personal development option.
Rank 6 in Agent (Law Enforcement) or Citizen (Corporate) and Rank 5 in Merchant (Corporate) can give a slight boost of +1.
Noble, Agent, Entertainer, or Scholar can get +SOC as a mustering out benefit.
Military (Army, Navy, Marines) will put you near the top if you reach the upper officer ranks.
Citizens (Colonist/Worker), Drifters, Scouts, Rogues, Merchants (Broker, Free Trader), Military (Enlisted) have no ability to improve their social status.
What's interesting is that the big money careers (Corporate Merchant, Broker, Corporate Officer) can barely shift the social scale (and only if you reach the pinnacle of your profession).
(Also, amusingly, Entertainers and Scholars have the best chances of improving SOC but at the cost of actually getting good at their jobs (aka rolling on +SOC on Personal Development instead of getting a skill

)
Also note that of those careers that can give you more than minimal SOC improvement, many are gated by SOC checks before you get those chances. Military Officer is SOC 8+ roll to enter (though Mil Academy does not have a SOC check gate), Corporate Official & Artist are SOC 6+ for survival every term, Noble is SOC 10+ to get into (if not already 10+).
All in all, Social Mobility is extremely limited and business success (aka money) is essentially irrelevant. It is interesting that scientists can be upwardly mobile.