You're assuming that the second crime is violent, which is curious given your assumption that a large fraction of prisons are in for non-violent crimes.

There's also the problem that convictions and sentences understate the crimes committed by those convicted because of plea bargaining. In other words, the fact that someone was sentenced for a non-violent crime does not imply that they didn't commit a violent crime.

And then there's the whole "gateway" theory.

I don't know how all these things work out - I'm just pointing out that the plausible theories go all directions.