I suggest that you concentrate on getting the sp working fast on its own without worrying about SSIS for now.
You should try to work out where the bottleneck is: is it in the selection of data from the main table, or in the writing of data to the staging table?
If you are writing millions of rows to the staging table, you may improve performance by writing them in batches - say 100,000 at a time.
Phil
If you haven't even tried to resolve your issue, please don't expect the hard-working volunteers here to waste their time providing links to answers which you could easily have found yourself.