The best answer is not in FATE, but in a Gumshoe game!
Gumshoe is a tabletop roleplaying game designed with the focus on investigation; finding out whodunit in a Gumshoe campaign is given more or less the same attention as how you go about stabbing goblins is given in a D&D game, and I've found it's wonderful at letting mysteries unravel. The idea I found especially useful was in a variant of Gumshoe called Night's Black Agents, and it works something like this.
Imagine you were about to prep a dungeon for your prototypical group of kick-in-the-door murderhobos in D&D. You need rooms, and you need hallways connecting those rooms to each other, and you need an entrance, and you ideally want to have things get harder as you move through the dungeon until a boss. You probably have multiple branching paths through the dungeon. Some doorways are hidden, but most are easily spotted once you have a moment to look around the room. Each room also has stuff in it; maybe traps, maybe monsters, maybe treasure, maybe a bit of all three. Go ahead, sketch a quick little dungeon on a napkin or something.
Alright, now take all those boxes you drew for rooms and make those into places that could hold pertinent information to your mystery. A grisly murder in the local diner, a sketchy corporation's main office after dark, the automotive chopshop run by lycanthropes. Not all of these places will have useful info, just like not all rooms in a dungeon contain treasure! What all of these places do have in common is that they contain clues leading to other parts of your mystery map. (Just like each room in a dungeon contains doors that lead to other rooms. On your map, mark each connecting hallway with the clue(s) that lead someone from scene to scene.)
The entrance room of your dungeon? That's the opening scene of your story. The monsters in the room? That's the mob thugs waiting to kneecap do-gooders. The treasure? That's probably still loot. (Don't mess with perfection.) The last room is, of course, the final culprit of the mystery, who will likely provide your climactic scene when you point the accusing finger at them.
The vitally important thing to remember is that most doorways in a dungeon are not hidden. Look at the clues, and hand them cheerfully to whoever has the most appropriate skill for noticing that clue. This is still FATE, so you can tell the players "The corpse has had their canines removed and that points to werewolfy things" before asking "so how do your characters notice that?" You can also hint at it and tell the first player to ask questions about something related. You do not want these clues hidden. (A dungeon crawl where we run out of doors two rooms in is lame, and so is a mystery story where the trail goes completely cold two scenes in and never picks up again. It's a good habit to keep at least one floating scene on hand for such an eventuality.)
I'm going to delve into that floating scene idea for a moment. Lets say the mystery the players are investigating was secretly a bunch of fallen angels trying to steal a macguffin. If I were running that game, I would have a bunch of thieves hired to steal the macguffin that can show up if the players lose the trail, acting suspicious and therefore giving the players a way back in. Those thieves could show up at any point in the mystery they're needed, or indeed not at all if they aren't. Still, if you planted and then revealed your clues right you shouldn't need this too often.
So yeah. Gumshoe is cool, Night's Black Agents is awesome, and while both are great extra reading for running investigation games I find the conceptual framework above improved my mystery campaigns by orders of magnitude.