After Aleinu L’Shabe’ach there is no need to recite another Kaddish Yehei Shelama because it was already said after Shir Shel Yom. Likewise, there is no reason to recite Kaddish on verses of Scripture twice within such a short amount of time. Even according to the Kavanot HaAri, there is no room for another Kaddish, and that is the custom of the Sephardim.
Nevertheless, in Ashkenazic minyanim, mourners are accustomed to reciting Kaddish Yehei Shelama on verses of Scripture twice. In other words, Kaddish Yehei Shelama is recited after Aleinu L’Shabe’ach and again after Shir Shel Yom. This law depends upon whether or not it is permissible to recite additional Kaddishim.
The poskim write that every person must hear seven Kaddishim every day, corresponding to what is written (Psalms 119:164), “Sheva bayom hillalticha” (“I praise You seven times daily”) (Beit Yosef 55:1; Mishnah Berurah 55:5). According to the Ari, one must hear twelve Kaddishim daily.
. Many Acharonim (among them, Knesset HaGedolah, Chayei Adam, Shetilei Zeitim 55:9) write that just like it is best to minimize the number of berachot that one recites, so it is best to say as few Kaddishim as possible. The Mishnah Berurah 55:1 adds that some Acharonim forcefully challenge the practice of those who assembled to recite verses or rabbinic teachings and say Kaddish a few times. Instead, one Kaddish is recited on the verses and one on rabbinic study, and no more. That is what is written in Az Nidberu 13:33, and hence Kaddish Yehei Shelama may not be recited twice at the end of the prayer service. By contrast, Eshel Avraham Butshatsh 132:2 writes that many Kaddishim may be added, and that the law concerning them resembles the law of the Thirteen Attributes of Mercy which are recited many times in Selichot. Since Hashem’s Name is not mentioned in the Kaddish, the Kaddish is not considered to be recited in vain. He continues that this is how he practiced when he was a chazan and there was no mourner present – he himself recited Kaddish Yehei Shelama twice at the end of the prayer service. The Ari’s opinion, that there is no place for a Kaddish after Aleinu, is brought in Kaf HaChaim 55:1; 48:1, at the end of s.v. “V’Da Hakdamah.”
Further, the Eshel Avraham 132:2 writes that it is proper to adjoin Barchu to Kaddish just as it is customary to do regarding the Barchu before Birkot Keriat Shema in Shacharit. Therefore, the minhag (in Nusach Sephard) to say a few verses before Ma’ariv and Kaddish after them became widespread. Similarly, at the end of the prayer service, it is proper to recite Kaddish before Barchu.
Moreover, formerly, the custom in Ashkenaz was that only one person would recite Kaddish, and when there were a few mourners they would take turns. There were times that every mourner merited reciting one Kaddish per week. Perhaps that is the reason that it was customary to say Mourner’s Kaddish twice, so that more mourners could recite it. When the number of mourners increased as a result of the pogroms, a new custom was introduced – that all the mourners would recite Kaddish together. By contrast, the Sephardic custom has always been that all the mourners recite Mourner’s Kaddish together.
. The following are the seven Kaddishim (as written in Mishnah Berurah 55:5): 1) Half-Kaddish after Pesukei d’Zimrah, 2) Half-Kaddish after Shemoneh Esrei, 3) Kaddish-Titkabal after Kedushah d’Sidra, 4) Yehei Shelama after Aleinu L’Shabe’ach (according to Nusach Ashkenaz), 5) Half-Kaddish after Ashrei in Minchah, 6) Kaddish-Titkabal after Minchah, and 7) Half-Kaddish in Ma’ariv between Birkot Keriat Shema and Shemoneh Esrei. The Kaddish after Ma’ariv, even though it is a mitzvah, is not included in the seven.
The Kaf HaChaim 55:1, based on the Ari, adds another five, as follows: 1) Kaddish d’Rabbanan before Hodu, 2) Kaddish d’Rabbanan after Pitum HaKetoret in Shacharit, 3) Half-Kaddish before Barchu of Ma’ariv, 4) Kaddish-Titkabal after Ma’ariv, and 5) Kaddish Yehei Shelama after the verses following Ma’ariv. Some are accustomed to adding another Kaddish on the verses recited after Minchah, which makes a total of thirteen Kaddishim. The Kaf HaChaim 55:21 writes further that if the mourners are minors, or the mourner stutters and cannot pronounce the words properly, it is necessary for another person to say Kaddish with him, so as to complete the required number of daily Kaddishim. However, regarding an optional Kaddish, not included in the twelve mentioned above, a minor may recite it himself (ibid., 19).