I'm reading all the Lee Child novels and am on my last one (
Running Blind) -- have NOT read them in order. I would say they fulfill your requirement -- something happens immediately at the start of the novel and then the hero, Jack Reacher, gets pulled into solving the crime. He's a loner who travels only with the clothes on his back and a fold-up toothbrush. Here's from Wikipedia about that particular title --
The story begins in New York, with Reacher confronting and beating up two thugs sent to collect money from the diner he was eating in as part of a protection racket, implying that he is a member of a rival gang. Reacher is then pulled up by the FBI and questioned over who he is working for, but explains he is a loner and has been since he mustered out of the army. He is then questioned about two women whose cases of sexual harassment he dealt with when he was an MP and it is revealed they have both been killed in the last few months and a criminal profiling team have come to the conclusion that the person who did was someone exactly like Reacher.
Here's an article about the Jack Reacher titles and a short summary of each (or click each title for a longer discussion) --
Jack Reacher - Wikipedia, the free encyclopedia
The authors I starred in my previous post usually start with a crime and then move into the story to solve it.