When my return has been rejected due to a SSN conflict, are there any additional forms I will need to include in my paper filing packets?

If your return was rejected because of a mistake made in your SSN, then you don't need to do any additional steps, just correct the mistake.

If someone stole your SSN to file the return, you should call the IRS Identity Protection Specialized Unit at 1-800-908-4490 for assistance and you'll have to complete Form 14039. 

After you complete Form 14039, mail it to the IRS with a copy of your Social Security card and driver’s license. If you don’t have a driver’s license, you can substitute a U.S. Passport, military ID or other government-issued identification card.

You can find more information here:

What to Do if Someone Has Already Filed Taxes Using Your Social Security Number