You must run high fever because of the cold. If so, you shall try to cure your cold first and for the most not just treat red eyes, because the bloodshot eyes are caused by the high fever of cold. So, take some medicine to treat your cold first. Or visit an eye doctor and ask him to help you treat cold as soon as possible. After you get recover from old, your eyes will be Ok and the redness will disappear.
This kind of situation is mainly caused by eye infection. Generally speaking, conjunctival blood vessels are in the ''sleeping'' condition and does not have blood through them. So the sclera is white, but once suffered invasion by bacteria and virus, the "sleeping" blood vessels will soon be ''wake up'', then expand. So red blood flows to the eye, and sclera become red. I suggest you continued to cure your cold, you can use some anti inflammatory and take more vitamin C. At the same time, you should use some eye drops prevent conjunctival infection.