Does she immediately go to the Beech Tree?

Go to a different tree, first

Hermione sprinted to the Whomping Willow. She knew if the Slytherin girls had a moment, they'd look through her bag and steal the cloak. She couldn't risk that.

Using a long branch, she poked the knot at the base of the tree, stopping its branches from swaying. She ran to the base of the tree and quickly shoved the folded cloak in the tunnel.

Snap. Hermione turned around quickly. She thought for sure she heard a branch break. She was scared that the tree had woken up earlier than normal, or worse, someone saw what she was doing. But she couldn't see anyone. The wind rustled through and bushes through the grounds, as she heard the distant sounds of the train in Hogsmeade. She'd have to hurry.

She got up and began to make her way towards the lake again.