First, it is important to understand the halacha, which you correctly assume, that one who puts on a tallit does not make a beracha when putting on his tzitzit in the morning.
The Shulchan Aruch[2] says that one whose hands are still unclean from the night should put on his tzitzit without a beracha, which he will recite later. He suggests that one recite the beracha while purposely handling the tzitzit or when he puts on another pair of tzitzit (i.e., a tallit). The Darchei Moshe[3] extends this by making it a regular practice – one makes a beracha only on the tallit that he wears for Shacharit, and this also covers the smaller pair of tzitzit that he wears under his clothes.
The Mishna Berura[4] reports that this was the minhag in his time as well, and he provides reasons for it. One is to avoid making two interchangeable berachot[5] in close proximity. Since one beracha can service multiple tzitzit garments, a second beracha on a tallit would be a beracha she’eina tzricha (an unneeded and thus improper beracha).[6] The Darchei Moshe[7] is bothered by the possibility that the tzitzit garment will be too small for the mitzva of tzitzit (and thereby will not warrant a beracha). There are other factors that could make a beracha on the tzitzit unnecessary (e.g., the garment’s shape or having worn the tzitzit all night[8]).
However, the common practice does raise problems. Berachot generally should precede the mitzva’s fulfillment; here, the beracha is made after the mitzva of tzitzit was performed. The Beit Yosef,[9] based on Rabbeinu Yona, explains that it is sufficient that the beracha precedes part of the performance of the mitzva, and in this case, the tzitzit continue to be worn after the beracha. Alternatively, the Taz[10] answers that since we assume that one’s hands are dirty when it is time to put on the tzitzit and he cannot make the beracha right away, it is appropriate to delay the beracha.[11]
Is this system best even if one will put on his tallit only a significant amount of time later – in your case, during chazarat hashatz? Here, the concern regarding the unnecessary recitation of two berachot in succession is reduced, and the problem of delaying the beracha on the tzitzit increases. Several poskim therefore say that when a long time is expected between the two, one makes a beracha first on the tzitzit and later on the tallit.[12] Some still prefer one beracha, recited on the tallit, because of the lingering concern that the tzitzit may not warrant a beracha.[13] This position is far from clear; note that when one does not wear a tallit, he takes his chances and makes a beracha on the tzitzit. The preferable practice should also depend upon whether the specific tzitzit garment definitely requires tzitzit or not.[14] However, it is difficult to alter minhagim.
It is also not clear what constitutes a long break. There appear to be different opinions, ranging from around an hour to two or three hours.[15] Therefore, if the gap between tzitzit and tallit is from the time one dresses until chazarat hashatz, there is ample justification either to make a beracha on each or make the beracha only on the tallit (if it is his own tallit or he acquires it temporarily before putting it on[16]). One can continue to act as he was taught or how he has practiced until now.
Either way, you should have the proper intention. If you make a beracha when putting on the tzitzit, intend not to include the tallit with that beracha; if you do not make a beracha on the tzitzit, have in mind when you say the beracha on the tallit to include the tzitzit, as well.