Beged Isha/Ish when no intent to cross-dress
You’re out for a walk with your spose or daughter and she is cold. You are wearing a men’s jacket and don’t mind giving it to her to wear. Is it permitted to do so, or does it involve the prohibition of Beged Isha?
Wearing clothing of the opposite gender due to the cold, or for purposes of shade or Tznius:
Many Poskim rule the prohibition of Lo Silbash only applies when one wears the opposite gender’s clothing for the purpose of resembling the opposite gender. It is however permitted for one to wear the clothing of the opposite gender for other purposes, such as to protect one from the cold, or from the sun [or for other purposes]. Other Poskim however rule that doing so is forbidden. [Practically one may be lenient in this matter in a time of need.] This however only applies if one’s gender is still recognizable. It is however forbidden in all cases to wear clothing of the opposite gender in a way that his gender is no longer recognizable.
 Shach 182/7 in name of Bach; Taz 182/4; Avnei Tzedek 72; Shraga Hameir 7/124 that the Shach agrees with the Bach on this matter; See Darkei Teshuvah 182/9; See also Admur 301/6 based on M”A 301/16 and Gemara Shabbos that a female ring is not considered a Shinuiy for a man to carry through wearing on his finger on Shabbos being that at times during the week they wear it on their finger to bring it to be fixed. Thus we see that if one wears the item for non-cross-dressing purposes, it is permitted
 The reason: As the verse states “Eideiy Isha” which refers to ornaments that beautify a woman and make her appear like a woman. [see Taz ibid]
 Yad Haketana 2/6 [p. 279], brought in Darkei Teshuvah 182/9; Binas Adam 90/94; The Shach ibid questions the ruling of the Bach, although some write that his question is not relating to this matter, of which he also agrees; See Divrei Chaim 2/62 [prohibits even to prevent pain]; Maharsham 2/243; Yabia Omer 6/14-4
 Shraga Hameir 7/124; Maharsham 2/243 rules one may only be lenient if it is recognizable to the viewer that one is doing so due to the cold and the like; Bach ibid writes one may be lenient because one has no other choice; See Minchas Yitzchak 2/108-4
 Shach ibid