The Kaddish that is recited after kri’at haTorah is connected to it. Therefore, there should not be a serious break between the end of kri’at haTorah (and its normal concluding beracha) and Kaddish. However, we have to look for precedents to see whether the recitation of HaGomel and the corresponding response constitutes a problematic break.
The Kitzur Shulchan Aruch[2] says that if the chazan at Mincha stopped between the end of Ashrei and Kaddish to put on a tallit, he should say a few p’sukim before Kaddish. This is because Kaddish relates to the p’sukim of Ashrei and because putting on the tallit is too significant a break. Following this approach, one would assume that HaGomel is also too great a break between kri’at haTorah and Kaddish.
We can counter this indication in several ways: First, there are other breaks that are not deemed problematic. For example, we do not recite Kaddish immediately after kri’at haTorah of Mincha on Shabbat, so that it will be recited directly before Shemoneh Esrei. That Kaddish, though, also relates to kri’at haTorah. The Magen Avraham[3] explains that hagba-gelilla and the p’sukim recited upon returning the sefer Torah are not considered a break. Nevertheless, we cannot use this as a firm proof because, as he explains, those parts of the tefilla are considered just an extended ending of kri’at haTorah. One can claim that, in contrast, HaGomel is unrelated to kri’at haTorah and constitutes a halachic break.
Second, returning to the first analogous case, many poskim, including the Mishna Berura,[4] say that putting on a tallit or tefillin is not a sufficiently long break to mandate repeating p’sukim before Kaddish. Since he stresses the break’s brevity, not its subject matter, HaGomel, a short beracha, should not be considered a break either.
Third, normal interruptions do not count as halachic breaks between Kaddish and the preceding passages to which it applies. For this reason, we say Kaddish Shalem, which relates to Shemoneh Esrei, despite the breaks for Hallel, kri’at haTorah, etc. in between Shemoneh Esrei and that Kaddish.[5] One can argue that since HaGomel is normally said at the conclusion of one’s aliya, it is, at least informally, part of the kri’at haTorah procedure and hence not a halachic break.[6] Still, one may counter that Kaddish Shalem is different because it was originally intended to be said long after Shemoneh Esrei. In contrast, the Kaddish after kri’at haTorah can, and perhaps should, be recited immediately after the end of the last aliya. Nevertheless, the idea that normal procedure does not constitute interruption likely remains pertinent.
Another difference between the cases is that the ba’al koreh, who usually recites the Kaddish, is not the one who is reciting HaGomel. In a similar situation, the Mishna Berura[7] urges the chazan not to talk between Shemoneh Esrei and Kaddish Shalem, to avoid an unwarranted break. However, we do not find that the rest of the congregation has the same restriction. Similarly, what the oleh does should not be so important. On the other hand, an opposing view can be taken, namely, that the whole congregation responds to HaGomel, and that public interruption is more problematic than an individual talking before Kaddish Shalem. Nevertheless, the fact that the ba’al koreh does not recite HaGomel seems significant, at least if he does not respond.
After comparing our case to halachic parallels and making distinctions, we conclude as follows: All things being equal, it may be preferable for the third oleh to wait until after Kaddish to recite HaGomel. After all, there is no halachic requirement to connect HaGomel to an aliya; indeed, one who says HaGomel does not need an aliya. However, if he wants to recite it before Kaddish, we do not have sufficient grounds to disallow him from doing so.