The custom of wearing a hat and jacket:
A. Custom, source, and reason:
The custom of wearing a hat over the Yarmulke: It is customary amongst pious Jews to walk with two head coverings, and they thus wear a hat over their Yarmulke. This custom is followed by Chassidim.
The Talmudic source of this custom: The source for this custom is from a Mishneh in Tractate Shabbos which states that that if Heaven forbid there is a fire in one’s house on Shabbos one may only save the 18 articles of clothing which he normally wears during the weekday. The list of clothing mentioned there includes “one’s hat, and the head covering under it.” Likewise in Tractate Chulin [138a] it states that the High priest wore a wool head covering under his hat. [Furthermore, some learn that the statement of the Gemara in Shabbos 118b regarding not walking with one’s head uncovered is actually referring to the second covering over the Yarmulke, and it is saying that one is not to walk without a hat over his Yarmulke.]
The inner meaning behind wearing a hat over one’s Yarmulke?
- A hat represents an Oar Makif. Thus, one who desires to completely accept upon himself the yoke of heaven is to cover his head with two coverings which represents two Makifim.
- Every Jew has two intellectual aspects, and the two coverings represent the higher fear and the lower fear which is drawn to him.
- Walking with a hat over the Yarmulke is a Segula for both spiritual and physical protection.
B. The specific times for hat wearing:
Wearing a hat during Davening: According to some opinions, during Davening, it is an actual obligation to wear a hat over one’s Yarmulke.
Wearing a hat during Bentching: A G-d fearing Jew is to wear a hat while reciting Birchas Hamazon.
Wearing a hat while reciting Hashem’s name in a blessing or prayer: It is proper to wear a hat over the Yarmulke any time that one recites Hashem’s name, such as in a blessing, or prayer. [For this reason, some wear two Yarmulkes at all times in order so their head be always doubly covered when they need to say a blessing. Others have the custom of placing their hand over their Yarmulke when reciting a blessing and don’t have their hat available.]
Wearing a hat during Torah learning: Some are particular to wear a hat over their Yarmulke even while learning Torah.
Is a Yarmulke which has two parts [inside and outside] considered a double covering?
It does not suffice as two coverings and so is the custom.
C. Wearing a jacket:
It is customary amongst G-d fearing Jews to wear a jacket or frock coat over their clothing even during the summer. The reason for this is because the inner clothing represent an Or Penimi while the outer clothing an Or Makif.
 Rav Yaakov Emdin in his Siddur, brought in Ketzos Hashulchan 3:4; Damesek Eliezer 6:7 writes that the custom of all the Geonim and Tzaddikim is to walk with a Yarmulke under their hat and that this is an age old Jewish custom; Mahritz Shabbos 118b; Maharsha Shabbos 156b; Shulchan Hatahor Mamar Tznius 4
Making the Yarmulke visible under the hat: There is a tradition recorded in the name of Rav M.M. Rimnov to be particular to extend the Yarmulke below the hat in a way it is visible on the outside in order to distinguish ourselves from the gentiles. [Piskeiy Teshuvos 2 footnote 64]
 Rebbe in Igros Kodesh 18 p. 394, brought in Shulchan Menachem 1:6
 Rebbe in Igros Kodesh 18 p. 394, as explained in Shulchan Menachem 1:6 and Kovetz Yagdil Torah (Israel) vol. 4 page 1827.
 Chapter 16 Mishneh 4 as explained by the Bartenura
 Mahritz Shabbos 118b; Maharsha Shabbos 156b; The Mahrtitz on the Gemara in Shabbos 118b [regarding not walking with one’s head uncovered] explains that it refers to wearing a covering over his first covering, as this brings person to fear of heaven. So also explains the Maharsha on Tractate Shabbos 156b regarding the mother of Nachman Bar Yitzchak that she was careful her son be with a double covering. [Brought in the Sefer Amudei Arazim.]
 Sefer Hasichos 1947 p. 158
 Midrash Pinchas 31 [Pinchas of Koretz]
 Tradition brought in Piskeiy Teshuvos 2 footnote 64
 See M”B 91:12; Piskeiy Teshuvos 91:3
 Admur 183:6
 Salmas Chaim 181; Sdei Chemed Mareches Brachos 1:42
 Perhaps this serves in the place of a covering just like was ruled in Basra 2:6 regarding placing one’s hand over his head if he does not have a Yarmulke. Now although doing so is invalid for reciting blessings, nevertheless since one’s head is already covered perhaps placing a hand over the Yarmulke serves as a “Heker” in place of the hat and is hence valid just like it is ruled in 2:6. Vetzaruch Iyun as to if there is any Rabbinical authenticity to this custom.
 Shulchan Hatahor Tznius 4
 See Ashel Avraham Butchach 240, brought in Orchos Chaim 240:8; Kaf Hachaim 240:53; Taharas Yisrael 240:53; Sheyikadesh Atzmo 32:5; Pashut! As the sewing of the two parts turns them into one garment and people do not consider it two garments. Is a coat with a wool lining considered two garments? Furthermore, if it were considered two garments then it should be prohibited to remove and wear being that one is to avoid wearing and removing two garments at the same time. [See Halacha 6E] Furthermore it would be considered as two of the 18 garments that may be removed in case of a fire.
 Toras Moshe [Chasam Sofer] 4 Pesach; Torah Leshma [Ben Ish Chaiy] 369
 Toras Moshe ibid