On Shabbat morning, after completing the Torah reading, before the Maftir reading, Kaddish is recited. This is generally done after seven Aliyot – the minimum amount of Aliyot called to the Torah on Shabbat morning. A congregation may, however, call more than seven Aliyot on Shabbat morning, in which case the Kaddish is recited after the final Aliya, before the Maftir, the Aliya which repeats the last several verses of the Parasha.
However, according to Sephardic practice, which follows the view of the Ribash (1326-1408), there are occasions when a second Kaddish is recited after the Maftir reading. This happens on a Shabbat when a special Maftir is recited, such as on Shabbat Rosh Hodesh, or on Yom Tob, when the Maftir reading is the section that deals with the special sacrifice brought on that holiday. On a normal Shabbat, the last verses of the Parasha are repeated only because it would be disrespectful to the Torah scroll for somebody to be called to read the Haftara, the section from the Prophets, without also reading from the Sefer Torah. Since there is no actual requirement to read the Maftir section on an ordinary Shabbat, a Kaddish is not needed after the reading of Maftir. However, on a Shabbat when a special Maftir is read, the Maftir reading fulfills a specific requirement. The special occasion obligates the reading of these verses. As such, on these occasions, the Maftir reading constitutes an independent Torah reading, such that a separate Kaddish is recited.
The concept underlying this practice is that each section of the prayer service is concluded with the recitation of Kaddish. This is indicated in a responsum of the Geonim stating that Kaddish is recited after Pesukeh De’zimra; after the section of Shema, Amida and Vidui; and then after Kiddusha De’sidra (Ashreh and U’ba Le’sion). Each section is concluded with Kaddish, and thus, by the same token, each Torah reading is concluded with Kaddish. (This point is made by Rav Natan Ben-Senor, in Ner Sion.)
This applies, as mentioned, on Shabbat Rosh Hodesh and on Yom Tob. One Kaddish is recited after the reading from the first Sefer Torah, and then a second Kaddish is read after the Maftir reading from the second Sefer Torah.
If Rosh Hodesh Tebet falls on Shabbat, then three Sifreh Torah are read, as special sections are read for Rosh Hodesh and also for Hanukah. In such a case, if seven Aliyot were called to the first Sefer Torah, then three Kadishim are recited – one after the reading from each Sefer Torah. If only six Aliyot are called to the first Sefer Torah, then two Kaddishim are recited – one after the reading of the Rosh Hodesh section from the second Sefer Torah, and another after the reading of the Hanukah section from the third Sefer Torah.
On Simhat Torah, too, we read from three Sifreh Torah. From the first, we read the final Parasha of the Torah, Parashat Ve’zot Ha’beracha; from the second, we read the first section of Parashat Bereshit; and from the third, we read the Maftir, the section that deals with the special sacrifice offered on the holiday. Two Kaddishim are recited – one after the reading of Parashat Bereshit from the second Sefer Torah, and another after the reading of the Maftir. Kaddish is not recited after reading the first Sefer Torah so as not to make an interruption between the reading of the final Parasha of the Torah and the reading of the first Parasha. The Rabbis teach that such an interruption gives Satan the opportunity to prosecute against us, and so we proceed immediately from the reading of Parashat Ve’zot Ha’beracha to the reading of Parashat Beresheet, without interrupting even for the recitation of Kaddish.
Ashkenazim, following the ruling of the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), abide by a different view, and never recite a second Kaddish after the Maftir. The practice of the Sepharadim, however, follows the ruling of the Ribash, as discussed.
Hacham Bension Abba Shaul (Israel, 1924-1998), in Or Le’sion (vol. 2), rules that even if a congregation has only one Sefer Torah on a Shabbat when a special Maftir is read, a second Kaddish is recited after the Maftir reading. A second Kaddish is recited on these special occasions not because a second Sefer Torah is read, but because the Maftir reading constitutes a separate requirement. Therefore, it makes no difference whether the special Maftir is read from a different Sefer Torah or from the same Sefer Torah as the rest of the day’s reading.
It should be noted that when the Torah is read at Minha, such as on Shabbat afternoon, Kaddish is not recited after the Torah reading. The reason is that Kaddish is recited right after the Torah reading, once the Torah is returned to the ark, before the Amida, and this Kaddish covers the Torah reading, as well. This applies even on Yom Kippur, when a Haftara is read after the Torah reading at Minha (the Book of Yona). The Haftara is considered an extension of the Torah reading, and there is therefore no need to recite Kaddish after the Torah reading.
Summary: On a regular Shabbat, when the Maftir reading merely repeats the last several verses of the Torah portion, Kaddish is recited only after the completion of the Torah portion, and is not repeated after Maftir. However, according to Sephardic practice, when a special section is read for Maftir, such as Shabbat Rosh Hodesh, or on Yom Tob, an additional Kaddish is recited after Maftir. This applies even if a congregation has only one Sefer Torah and the Maftir is read from the same scroll as the rest of the reading.