With H-shem's help, we have begun learning Hilchot Tzitzit with the Ben Ish Chai, Hacham Yosef Hayyim of Baghdad. The sefer is translated by Shmuel Hiley and published by Yeshivath Ahavath Shalom Publications in Jerusalem 5765/2005. He calls the names of the Chapters by the names of Parashiot, so instead of Chapter One, the first chapter is called Parashath Bereshith, Laws of Ziziyoth. Please note these are for learning purposes only. For the halacha lemaaseh, i.e. for what to do yourself please ask your own Rav.
Parashath Bereshith, Laws of Ziziyoth, Page 7-8: This is a long entry so we will break it up over a few days, BE"H
10 (continued). If one puts on another tallith qatan, even immediately after taking off the first one, it is necessary to recite another berachah over the second one. Similarly, if one takes off one tallith gadhol and then puts on a second one, one must recite a new berachah over the second tallith gadhol.
In my work Rav Pealim (Vol. II, Orah Hayyim, chapter 5) I cite a query regarding the berachah over a tallith from someone who is accustomed to put on zizith and tefillin at home, in order to enter the synagogue whilst already wearing them. However, he is not able to put on at home the large tallith gadhol which he wears in the synagogue, because his way there takes him through busy streets full of people, Jews and non-Jews, and he does not wish to attract attention. Instead, he puts on at home a very thin tallith gadhol which, although of sufficient size, is not as large as the usual tallith gadhol, and can be worn under a coat without being conspicuous. Later, when he enters the synagogue, he puts on a full-size tallith gadhol, and leaves the smaller one under his coat.
I answer that he may recite a berachah in the synagogue over the second, larger tallith gadhol, even though he is still wearing the first, smaller one, because the way from his home to the synagogue takes some time, and thus constitutes an interruption - increased by the fact that, on his way out of the house, he recites various verses whilst standing near the mezuzah. Furthermore, after he puts on the full-size tallith gadhol, he wears it properly over his head and body, while the first tallith gadhol remains on his shoulders, under his coat. For these, amongst other reasons, I rule that a berachah may be recited without question.
This learning should be in memory of Maran, HaRav HaGaon Ovadia Yosef, ztz'l.