When a beautiful young woman is found brutally murdered, a detective must travel to the remote island to find the killer before they kill again!

Give the town's people a goal. Don't know if this is your story, just a suggestion.

a beautiful young woman is brutally murdered, a detective travels to the remote island to find that the population is secretive and obstructs the investigation at every turn.

Put some oppositional force. Or like suggested state a ticking clock leading up to the next murder.

What Nir Shelter and FFF said. There is nothing that stands out, that differentiates this story from all the other stories about detectives tracking down a murderer.

(And how does the detective know the murderer will kill again?)

Why a remote Island. Which Island?