New Position: Guard
OK, this one might be cheating a little bit, since Stingily is an unsigned rookie, but the Titans are no strangers to playing low-pick linemen early (Amano was a seventh rounder who played in 15 games his rookie year).
Stingily played tackle in college, but I think his place with the Titans is at guard. Stingily is a little tall for a guard, standing 6'5" tall, but with Tennessee's bookend tackles both standing 6'7" tall, I don't see his height being a problem for a line that's already tall. The reason I think he is built to be a guard more than a tackle is Tennessee's depth chart.
Tennessee has one of the best tackle combinations in the league with Michael Roos and David Stewart, along with very good backup Mike Otto and developmental tackle Troy Kropog, all of whom are pretty young. There are a plethora of guards/centers on the roster, but as is common knowledge, the Titans guards have not been that good as of late.
The Titans almost always take their linemen late in the draft: Roos was picked in the second, Harris in the fourth, Amano in the seventh, Scott in the fifth and Stewart in the fourth. Stingily, drafted in the sixth, would fit right in to the trend.