Yes, the unchecking of columns in first row will not help if you are putting data into a file for the first time. Search google for "ssis merge flat files". I found one solution here - http://social.msdn.microsoft.com/Forums/sqlserver/en-us/771a7e11-cf32-48af-bd64-dfc058819e19/how-to-merge-csv-files-using-ssis?forum=sqlintegrationservices
try it and see how it goes.
A foreach loop to iterate over each of your CSV files.
Inside the foreach loop, I would add a data flow. Then in the data flow create a single Flat File Source and a single Flat File destination. The destination should be setup to append, not overwrite. Then, you can configure the Flat File Source via expressions to take its current values from the Foreach Loop so that each time a new file is encounted, the data flow executes, appending data to a single file.
There are plenty of examples of how to do this on this forum and on the 'Net. So, for the time being, I'll leave that exercise for you to perform. Search for "foreach loop expressions data flow" and see what you come up with.