I thought I'd share what solved this issue.

In the Flat File Connection Manager Editor (r-click and select Edit on the Connection Manager itself, not the Flat File Source Data Flow Transform), under Advanced, you can change the size of the property 'OutputColumnWidth'.

For each column I set mine to 500 and I stopped getting truncation errors. I didn't know up until now that a precedent for the width of each column was set within the Connection Manager, or that it could be edited!.