Another possibility, also by Niven, is "Death by Ecstasy" (1969).
The protagonist, Gil Hamilton, works for a worldwide law enforcement organization known as ARM. Since Gil has a psychic third "arm," he has the nickname "Gil the Arm."
In this story, Gil is asked to investigate the death of a former crewmate of his, a Belter, who has apparently died of a wire addiction.
The story is set in a post national Los Angeles, but it's not post-apocalyptic.
There is an element of romance in the story, since Gil picks up a transplant surgeon named Taffy, who he maintains a relationship with through subsequent stories.
It was collected in The Long ARM of Gil Hamilton (1976), which still fits your timeframe, and may have seemed like a novel since the three stories (two novellas and a novelette) in the collection all featured Hamilton and had other continuing characters.