No, there does not always need to be a leap of faith.
Another way for a character to change is one where the character changes gradually over the course of the story without awareness of the change. We call it a creep instead of a leap.
One of the best examples I can think of is Hamlet in Shakespeare's "Hamlet." Hamlet starts the story as someone who thinks too much (Thought as MC Problem). When given knowledge (MC Solution and OS Solution) of his father's murder by his father's ghost, Hamlet's reaction is to think it away ("Maybe the ghost is really a demon from hell sent to trick me" [paraphrased]). Hamlet is supposed to reveal his uncle, the new king, as the murderer.
Fast forward to the end of the play. The last scene has Hamlet in a duel with Laertes. Hamlet is acting as the King Claudius's proxy (!) in the duel. When the Queen, Hamlet's mother, drinks wine poisoned by King Claudius, Hamlet does not give a moment's thought, acts on the knowledge and kills the king on the spot.
At no point does Hamlet make a conscious choice to change, though he is changed over the course of the story.
Case in point: One big debate through the years is whether or not Hamlet was crazy. When the ghost tells Hamlet about the murder, Hamlet's approach as a Be-er is to pretend to be crazy. Since he is changed over the course of the story in a 'creep', not a leap of faith, it is unclear if part of the change includes going from pretending to be crazy to becoming a nutcase. My vote is the former, primarily because of my understanding of the storyform for the story.