Can you give a bit more details on your system as well as the symptoms?
Stuff like your OS (do you have any service packs installed? etc.). Also which page do you get redirected to? Does anything else go weird in IE? (stuff like hyperlinks changing targets etc?)
CWShredder seems to be the program to use to remove COOLWEBSEARCH. I doubt that it is malware in itself, since the author have written other utilities aswell that is widely used in spyware removal.
When you scan with CWShredder, make sure that all browser windows are closed, as well as all Windows Explorer windows/folders. In fact, make sure that only the necessary processes are running. (Use the task manager (CTRL-ALT-Del), go to the processes tab, and end all the processes that you don't know what they do. (If its a system critical process, windows will not allow you to end it, so don't worry about that, or only end the ones thats running as your user name (you probably won't need anything more that explorer.exe to run))).
After that, run CWShredder again (making sure you have the absolute latest version).
The problem with CWS is that there are so many variants, and each one removal differs from the next. Check out these sites:
http://www.spywareinfo.com/~merijn/cwschronicles.html
http://www.doxdesk.com/parasite/CoolWebSearch.html
http://www.spywareinfo.com/articles/cws/
I do think however that you should consider changing your browser (I know this have been said countless times in the past).