If a reaction spell has material components, do I already need to be holding them to cast it?

Some spells that are cast with a reaction, such as Feather Fall, require material components. Suppose I am using a component pouch instead of a focus, but I’m not currently holding a small feather. If a trap door opens under the party, am I able to quickly reach for the feather in my pouch as part of the reaction to cast Feather Fall, or am I out of luck?

Obviously this is mostly a non-issue for a caster who uses a spellcasting focus instead of a component pouch, since they are pretty much always holding their focus, and I don’t think there are many (any?) reaction spells with costly or consumed material components.