I don't know of another place to get Winamp. You can hunt the software sites, but those would probably link you right back to Winamp's site.
You could keep trying to download Winamp at different times of the day/night.
Right-click and save-as?
Or you might try shutting off your firewall and antivirus. If you have a router, maybe you could DMZ that computer for a while (shut off all protection for that computer - Linksys calls it "DMZ").
(Your firewall could be keeping them from doing a "reverse DNS lookup" on you. That can stop things with some web sites, though they really should expect it, in this day and age of firewalls).
You could install Firefox, and see if that'll download it. Or, if you're running Firefox, you could temporarily try IE again (without making it default).
There's probably a timeout setting in each browser, that you could increase.
Or you could try a download manager...