The first CT scan was almost certainly what's called "CT-KUB" (CT of kidney, ureters, and bladder) which is a quick look for stones or blockages. Although quicker, an ultrasound look is less sensitive, especially when a suspected stone may still be moving (in the ureters from kidneys to bladder) as is often the case with abrupt new onset of pain. The planned second CT scan with contrast (called an "CT-IVP" or CT intravenous pyelogram) can detect stones not seen in a regular CT. It's also used to look for a pouch ("renal diverticulum") inside the kidney itself where products can gather and stagnate to create stones or infections. Because the kidneys can react (spasm) to the injected contrast material (newer dyes reduce this), some prior blood work (esp. creatinine) is needed to evaluate their condition.
The abdominal CT provides a wider look at organs and spaces beyond the KUB; from about mid-ribcage down to about the hip joints. Besides the major organs and vessels, abdominal lymph nodes can also be evaluated; these are involved with the immune response to infection, cancer, etc. Also, there is a phenomenon known as "referred pain" where the reported site of the pain is not the site of the actual problem*. Suspected stone pain may actually be a problem, for example, of a stomach ulcer, or gall bladder (cholecystitis), or intestinal problem (diverticulitis). An abdominal CT should help consideration of such possibilities.
*The most well-known example of referred pain is the left-arm pain sometimes accompanying a heart attack.
