I'm Doug, and I'm very sorry to hear of your situation. My goal is to provide you with excellent service today.
Unfortunately, the only way you could force your local Goodwill store to allow you to go through the unsorted donations is with an order of the court---and that is not something that you will be granted.
However, Goodwill knows that while a donation once made is a gift under the law, that when there is a bone fide error made by the person making the donation as to the identity of the item given, that the donor has the legal right to set aside the gift and reclaim the mistakenly donated item. Goodwill is very good about helping people recover unintended donations.
If I might make a suggestion, you might consider offering a reward to the manager of the store, agreeing to pay, perhaps $50 to the finder of the coat, and an additional $50 to Goodwill Industries when the coat is located and returned. That should have the effect of motivating the finder to come forward with the cost when it is located.
You may reply back to me using the Reply link and I will be happy to continue to assist you until I am able to address your concerns, to your satisfaction.
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I wish you and yours the best in 2016,