We do call it "khăn" for short, but it depends on the context of the conversation, too. For example, the rag used to clean around the house is also called "khăn" for short. Therefore, we teach "khăn quàng" to specifically point to the "scarf".
Just a side note, "quàng" itself is a verb, meaning "to put on". Before getting out of the house in winter, we remind each other: "Nhớ quàng khăn!", which means "Remember to put on the scarf!"
Also, just a fun fact: the red neckerchiefs that teenagers in communist countries have to put on is called "khăn quàng đỏ" in Vietnam (we have to wear it since Elementary School and throughout Secondary School), which literally means "red scarfs". You can read more about it here https://en.wikipedia.org/wiki/Red_scarf.
Enjoy learning Vietnamese!
If you point to it, or discuss about it, you don't have to mention quàng since people in conversation know what kind of Khăn you're talking about. But if there is no previous reference, you need quàng since there are several types of Khăn.
Most Vietnamese don't use Khăn quàng since it's hot as hell over there. You don't need to wrap your neck with anything, but you use a head scarf (khăn choàng đầu) to cover your head, which is pretty much identical to a scarf, just much thinner and lighter.