raincoat (spoiler)

Hire tells Alice he was watching through the window as usual when she hid the raincoat. Later he writes to the inspector that he himself had picked up the raincoat and then hidden it after Emile had thrown it away. No doubt Hire gives a different account to the inspector so as not to cast suspicion on Alice, and it is his statement to her which is the truth. But if Alice hid the raincoat, one wonders how Hire came by it. (The handbag was hidden in her appartment. If the dangerous evidence of the handbag was retained there, the raincoat might as well have been too. But Hire never entered her appartment.) More importantly, why would Hire want to have the raincoat, and why later leave it for the police to find? Surely he was more concerned about Alice staying out of prison than Emile going inside. But Hire's letter with the raincoat could only increase police interest in Alice (and in Hire). And if Emile was apprehended there would be a further risk of his implicating Alice. So Hire should have kept or better destroyed the raincoat and written no such letter. He was notably logical apart from his infatuation, and his infatuation would surely have steered him away from helping the police. No?

"I beseech ye in the bowels of Christ, think that ye may be mistaken."