I found this, from the Chicago Community Kollel. Footnote 8 does quote a heter. Ask your posek; I don’t know if it is commonly accepted.
Halacha Encounters
Switched Coats
Rabbi Moshe Rosenstein
The Halachic Background
The Gemora [Bava Basra 41a] and Shulchan Aruch [Choshen Mishpat 136:2] [clothing that was switched with his own]
The Practical Halacha
We will discuss how the poskim instruct us to deal with various different practical scenarios of switched articles of clothing.
An item is missing and there is another, similar item left in its place:
Even if one is completely certain that his item (coat, towel, hat, etc.) was taken accidentally and he is certain that the item left behind belongs to the person who took his (i.e. it is very similar and hanging in the same place his was left), under most circumstances it will not be permissible for him to take the item left behind, even for temporary use.
There is, however, an exception to this rule. If the finder can assume that enough time has passed and the true owner of the item left behind has already realized that he accidentally took the wrong item, it is permissible for the finder to take and use the item left behind.2
Some poskim, however, do not consider this a permanent status. They contend that if two criteria are met, the finder may indeed use the item he took home. If both
If these conditions are met, the poskim say, it can be assumed that the original owner has not only given up hope of finding his item, but he has also resolved himself to considering this switch permanent.3 Some add that before using the item the finder should estimate its monetary value and write in his monetary record books that should he find the actual owner he will pay him back for the item.4
There is, however, an exception to this leniency. If the switched items are exactly the same, then there is no way for the finder to ever be sure that the original owner has even realized that a switch took place. Under these circumstances, it will not be permissible for him to use the item he has found.5
A prevalent custom not to be particular about allowing use of an item
There are certain rulings found in the later poskim that should be mentioned regarding this topic. As with all issues of this nature, a shailoh should be asked if one feels that a situation has arisen that would require clarification of the Halacha.
Additionally, if the finder knows who the owner of the left-behind coat is, and knows that the owner would not mind allowing him to use his coat, it will be permissible.
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1 See Rashbam B.B. s.v. harei zeh and Shulchan Aruch C.M. 358:5
4 Mishpitei HaTorah vol. 3 3:17. [He does not mention the reasoning above in note 3.]
5 Pischei Choshen ibid. s.v. ela.
6 C.M. 136:2
Rabbi Rosenstein is a full-time member of the Kollel and is a frequent contributor to Halacha Encounters.