Delete Server Behavior works great!
The timing was perfect on this post as I needed to implement this very functionality today. It works great!
Two adjustments to be aware of:
1. Be sure that the code to delete the current file is above the Upload File code if you have a redirect page.
2. In the Delete File code, replace cStr(Request.Form("Upload")) <> "" with Request.ServerVariables("Request_Method") = "POST"
Thanks for the help Ray!