This sounds somewhat similar to a problem with the 5th Gen Camaros.
Since the SS and Camaro are based on the same platform they "might" share the same issue.
Here's the info from the Camaro TSB and I hope it helps diagnosing your issue:
2010 Chevrolet Camaro | Camaro VIN F Service Manual | Document ID: 2907839
#09-01-38-003C: Moisture from Center HVAC Vent when A/C is On, Front and Back Floor Carpet Wet (Realign A/C Drain Tube) - (Sep 12, 2012)
Subject: Moisture from Center HVAC Vent when A/C is On, Front and Back Floor Carpet Wet (Realign A/C Drain Tube)
Models: 2010-2013 Chevrolet Camaro
This bulletin is being revised to add the 2013 model year. The Correction has also been updated to provide better clarification on how to access the grommet. Please discard Corporate Bulletin Number 09-01-38-003B (Section 01 – HVAC).
Some customers may comment that when they have the A/C on, they feel moisture coming from the center HVAC vent. They may also notice that the front and back floor carpet is wet.
This condition may be caused by a collapsed A/C drain grommet (example shown above), resulting in a buildup of water in the HVAC module.
Remove the A/C drain pass-thru grommet and realign the drain tube using the procedure listed below.
Note: If the carpet is wet, it must be removed and dried.
Raise and suitably support the vehicle. Refer to Lifting and Jacking the Vehicle in SI.
Support the transmission with a suitable transmission jack.
Remove the transmission mount bracket (support) bolts (1).
Slightly lower the transmission. This will allow access to the HVAC drain tube and grommet.
Reach above the transmission on the passenger side for automatics (shown above) or on the drivers side for manuals until you feel the HVAC drain tube and grommet.
Remove the HVAC drain tube grommet. Let any/all of the water drain out completely.
Important: Only apply a small amount of grease. Too much grease will cause another restriction in the drain tube.
Apply a small amount of grease, GM P/N 12345579 (in Canada, use 10953481) around the circumference of the HVAC drain tube and grommet (1).
Important: When installing the HVAC drain tube and grommet, make sure they are properly aligned to allow the HVAC water to drain properly.
Reinstall the same HVAC drain tube grommet. Make sure the drain tube is properly centered in the grommet.
Raise the transmission and reinstall the bracket (support) bolts.
Tighten the bolts to 58 Y(43 lb ft).
Lower the vehicle. Let the vehicle run with the A/C on.
Perform a final inspection for the A/C and verify that the system is draining properly.