HaRav Yosef Dov Soloveitchik has ruled that one may take cold dry food which is fully cooked out of the refrigerator and reheat it on the blech on Shabbos, provided that this food was on the blech as Shabbos began and was first removed on Shabbos.
In this case, returning the food to the blech (Hachzorah) is simply a continuation of the activity known as Shehiyah, that is, placing food on the fire before Shabbos and leaving it there until needed on Shabbos, which is permissible if there is a blech (See Shulchan Aruch Ibid. Sif 1). Regarding the other three conditions cited above, the aforementioned Ramo (Ibid.) quotes from the Ran that the last two, namely still having the pot in one’s hand and having the intent to return it, are not required when the food is removed from the blech on Shabbos and is to be returned on Shabbos. It appears that the Vilna Gaon (Biur HaGra to Sif 5 Ibid. s.v. U’Bilvad) as explained by the Mishnah Berurah in the Biur Halacha (Ibid. s.v. U’Bilvad), holds that in such a case, even the other condition, that the food must still be hot, is also not required if it’s a dry cooked food.
HaRav Soloveitchik extends this view to allow reheating on the blech a fully cooked solid food even if it has been completely cooled off in the refrigerator, because this is still just a continuation of the original Shehiyah.
It is noteworthy that Rav Ovadyah Yosef (Sheilos U’Teshuvos Yechaveh Da’as Chelek 2 Siman 45) allows this as well, but for a completely different reason.