Question: I accidentally took someone else’s umbrella in shul leaving mine behind. Could I have used it again to bring it back to shul the next morning?
Answer: The Gemara (Bava Metzia 43b) teaches that there is a machlokes as to the status of one who borrows an item without prior permission. Elsewhere, the Gemara (Bava Basra 46a) teaches that one who took the wrong clothes from a craftsman in error may use them until they are exchanged with the rightful owners. However, one who took the wrong clothes home from a shiva house or chasuna may not use them.
The Rema (CM 136:2) adds that one must return the items that one took to the rightful owner even if one does not receive their own items.
However, the Aruch Hashulchan (CM 136:2) writes that the accepted practice in populated areas is that one who accidentally took someone else’s overshoes may use them in the meantime, until they are exchanged with the rightful owners. As this is the accepted practice, it is not considered to be stealing.
R’ Moshe Feinstein (Igros Moshe OC 5:9:7) extends this to coats and other items that people mistakenly take, and advises that shuls adopt this as a matter of policy, allowing people to use theirs if taken by accident. If it transpires that the owner did not take theirs, they must offer to compensate the owner for the use of their clothing (See Mishne Halachos 5:276; Shevet Halevi 6:238).
R' Moshe Sternbuch (Teshuvos Vehanhagos 1:818) writes that one who may use an umbrella that one accidentally took home. People are typically not too bothered about lending umbrellas out to others and are happy for it to be replaced if necessary (See Minchas Yitzchak 8:146).
In conclusion, one who accidentally takes another’s umbrella home may continue to use it until they find the rightful owner. They are liable for any damage that may occur, and to replace it if necessary.