Down the Rabbit-Hole
Suppose we want to convert Alice in Wonderland to an interactive format. Which actors, items, and locations play an important role? What does a map of the world look like? What challenges did the protagonist overcome?
Player (Alice) begins the game at the Riverbank. Rabbit enters the scene.
Rabbit leaves the area, heading toward the Field. Player is supposed to follow Rabbit (but can do whatever they like).
When Player and Rabbit are both in the Field, Rabbit exposes a Hole under a hedge, previously undiscovered. Player now sees that they can enter Hole.
Player enters Hole, scores some points, falls down shaft (point of no return). Rabbit is here; he heads off to Tunnels; player can follow Rabbit through Tunnels to Hall.
That was a good warmup for the player. Now, you've got a chance to take some liberties and make things more interesting:
There were doors all round the hall, but they were all locked; and when Alice had been all the way down one side and up the other, trying every door, she walked sadly down the middle, wondering how she was ever to get out again.
Instead of a strict interpretation of the original story here, this looks like a great chance for a hub. Rather than being locked, some of the doors could open onto locations from other parts in the story. Maybe the cake that causes you to grow isn't here, and instead you need to visit the caterpillar first.
Advice from a Caterpillar
Okay, I'm not a caterpillar. But here's some advice anyway, in no particular order.
Map out every location in the game, and decide which areas are locked until some task is complete. If there are some "points of no return," those could be good places to break between chapters. Make sure that the game can't be in an "unwinnable" state at the beginning of each chapter. Your chapters (and timeline) don't need to follow the flow of the original story. Let the player work on several tasks simultaneously, within each chapter.
Let the story tell itself incrementally, through the player's actions. Let the player discover things by examining other things. Let them get stuck sometimes, and let them get "killed" (maybe Alice is rudely awoken and it was all a dream).
After our linear intro and a non-linear hub or two, we arrive at the final trial. Here's your chance to put multiple (non-death) endings into play. All of the extra side-quests, secrets that you discovered, and out-of-the-box, non-canonical solutions to problems you figured out determine the jury's final verdict, and the game ends in victory.
"But wait, what about branching?"
I don't think we needed it. Our focus was on puzzles and progression. The bulk of the game still had a non-linear feel, even though it had a clearly-defined beginning and end. The player was allowed to explore the game world as they liked, for the most part, and interact with it through trial and error.
True branching story lines are pretty rare; there's usually basically one overall plot line, where you may get to do things in a different order, or make some choices that affect later parts of the story (good choices could unlock secrets, bad choices could make side-quests unavailable). If your goal is to make an enjoyable game without inventing an entire fictional universe, I think this approach will do the trick.