My flat file connection manager has no file name selected because it's inside a foreach loop. With no file name selected, I can't add any new columns. These questions don't have the "no file name" issue, so they didn't work for me:
Add extra external column to flat file source
How to change flat file source using foreach loop container in an SSIS package?
How do I add a new column to my file format?
Here are my flat file connection manager and foreach loop: