Are there multiple files in the source? I'm not much of an SSIS guy, but the pattern I've seen is use a loop to process all files in a folder, move, load, ignore, whatever. Then as you process each file, you have the name of that file inside the package and can access that. Usually people loop through the files, do something, then move. You're the opposite in that you loop, move, then load. I'd think you really are looping, move, then another loop (perhaps another package) that processes the folder in which you've moved files.

If you process the folder and enumerate each file, the file task inside your loop should be able to access each name and load that.