More or less any time Annie Hacker appears, there's a 50-50 chance of Jim proving his humanity in a way most life-affirming.
Jim getting tearful when he gets it into his head that Sir Humphrey is dying.
When Humphrey puts down the phone at the end of "Party Games", Hacker, in an agony of uncertainty, asks him "Is it...?" and points more or less at himself. Humphrey, who has been stone-faced throughout the phone call, suddenly smiles at him like a proud dad, and says respectfully "...Yes, Prime Minister." Turns into a Crowning Moment of Funny as Hacker, feeling destiny call, stares nobly into the middle distance and unconsciously slips his hand inside his jacket like Napoleon.
When Sir Arnold is able to prove Humphrey innocent of espionage in "One of Us", Jim is driven to tears of joy.
One for Bernard in the final episode 'Tangles Web'. Having found out about Humphrey's indiscretion whilst being interviewed at the BBC he uses his own initiative to convince the producer to give him the master tape. Instead of simply giving the tape to Humphrey and getting his patronage for the rest of his career he chooses to give the tape to Hacker so that Hacker can use it as leverage to get out of a sticky situation Humphrey put him into and bernard feels sympathy for Hacker. Given how self serving civil servants are played throughout the series it really is touching to see that there is one civil servant who has a conscience.