We know that there is no prohibition of cooking something that has already been cooked, and we also know that this Halacha is only regarding dry and solid food. but we cannot rewarm liquid food. What is the Halacha if the food is congealed but will liquefy when reheated?
As mentioned in the question, the Shulhan Arukh (סימן שיח סעיף ד) rules that if a cooked liquid has cooled down to below yad soledet bo, (the definition of יד סולדת בו is whatever is between 45°C/113º and 71°C/160ºF). Since we do not know the precise temperature of yad soledet bo, any heating in that range may be considered bishul. But one may heat outside of this range.
For example, if water has been heated to 71°C/160ºF, one may then put the water in a place where it will get hotter, because of ein bishul ahar bishul. Similarly, one may put cold water in a place where it will heat up, as long as the water does not exceed 44°C/110ºF, because this kind of heating does not qualify as bishul.) it is no longer considered cooked, and it is prohibited by Torah law to reheat it past yad soledet bo.
Therefore, Sephardim may not take soup out of the refrigerator and heat it. Similarly, if they remove soup from the blech, and it cools off to below yad soledet bo, they may not replace it on the blech. If it is known with certainty that the liquid is still yad soledet bo, it may be replaced (as long as it meets the conditions of hazara). If one had situated it near the edge of the blech (which tends to be cooler), he may move it to the center and thus heat it more effectively.
If food is congealed but will liquefy when reheated, like mushroom gravy, it is considered a solid food, and one may heat it (כף החיים). However, according to Ashkenazic practice, le-khathila one should not heat it because some maintain that, le-khathila (i.e. from the start). One should not take an action that will cause food to change from liquid to solid or the reverse because this is like producing something new. However, if the liquid is secondary to the solid food (such as gravy for meat), then Ashkenazic custom would permit le-khathila heating the food together with the congealed sauce or gravy even though that might cause that sauce or gravy to change from a solid to a liquid.