SSIS add column from variable to flat file source

I am importing multiple flat files in SSIS and inserting the data into a table then I update a column in the table with the file name that I imported the data from.  This works fine but I think the update is making it take a LOT longer then needed.

I want to be able to set the value of a field in the destination source to a value from variable (the file name) at the same time I insert the data from the flat file source so I do not need to do the update after the fact.  I think this would save a lot of processing time.

I must not be searching Google for the proper terms to find a way to do this as I am sure something similar is being done all over the place I just can not find a way to do it.

If you need more details let me know.