Nobody knew exactly how to stop the Calamity until they found the
tape which explained that it can only be stopped if the Extra is killed (send back to death).
So there was not really the need for Misaki to tell anyone.
Other than that, she didn't tell anyone because no one talked to her except Kouichi.
And she couldn't tell him since
Kouichi and Reiko were related (his aunt).
So he would have had doubted Misaki.
Lastly, if she told it from the start, the producers couldn't have made anyone wonder who the Extra is. So it's part of the story.
In addition to this, a comment on Wikia about Class 3-3 might be even better in explaining it:
Katsumi Matsunaga's tape, about "sending the dead back to death",
she thought knowing was irrelevant. She knew, but what could anyone do? She held back until the end of episode 10, "Glass eye", apparently because it would have been rude to tell Kouichi that
the extra was someone close to him
and Kouichi didn't know of
her ability to see the color of death
until almost the end of episode 10. Her statement naming the extra was interrupted by Naoya Teshigawara. It wasn't until the end of the next episode that she determined to go, by herself,
and kill the extra.
-- And worth noting that the removal of the extra doesn't immunize the class from death, anymore than no historical calamity immunizes the other classes from ordinary accidental or illness death."