Deleting Rows is Increasing Index Size
Alright so I believe you doing and INSERT is increasing the size then after you do your OLEDB command which is sending hundreds of thousands of DELETE commands to SQL Server is what's causing your problem. So here is a short Demo/Tutorial on how to do this DELETE operation more effectively. Even if this doesn't solve your index problem it will make your package run more efficiently. 

STEP 1: Build a staging table for this information.

You're going to want to CREATE a table that's sole purpose is to hold the columns necessary for the delete. Whatever makes up your Primary Key will do. 

STEP 2: Truncate the Staging table. 



It's important to truncate first as this will all for re-run if the package fails.



Step 3: Data Flow task.

Next add a Data Flow Task and open. 



If you are doing your archive elsewhere than disregard the Multicast and the Archive destination in the next image.



STEP 3: Delete 





That should give you a pretty good idea of how to do this if you have any other question feel free to ask.

The reason this will be more efficient is because you are then doing set based operations instead of sending 350K DELETE commands to SQL server.