Sounds as though you're setting the destination at schedule time to something other than 'Default', then specifying the file name. I have a report that is run multiple times every month with different parameters, specifically different customer numbers, and it is written to unmanaged disk with the file name reflecting the customer number/name in the file name.
The only way I could find to accomplish this other than the brute force way that you're probably going about it -- manually keying in the parameters AND the destination information - was using a third party tool call APOS InfoScheduler which allows you to set up an Excel spreadsheet with the schedule info such as parameter values, destination, format, etc. You can then save the spreadsheet and change the parameters as necessary without having to rekey everything. You can even use formulas to calculate values such as the file name based on parameter values like customer number or name, date range, or other parameter values. As long as you have a reasonable command of Excel formulas, this is fairly easy to do.
Once you've got the spreadsheet set up, you just click one button in InfoScheduler to submit the jobs to the APS. It may be worth looking into.
--Bearden