Unfortunately, starting the clock on that second contract doesn't matter if you can't play out your first. I'd like to see some numbers on how many of these late round/undrafted early entries get second contracts.
I'd guess the numbers are pretty bad; I think they're bad for all rounds and probably worse for late rounds.
Personally, I think White is the only player who should be leaving early. He is the only player guaranteed to be a probably first rounder, if not a definite high second. Everyone else will probably be third or lower. I've always understood why players who are first rounders want to leave. Also if you have family issues or just can't cut it in school. However I've always felt it's better to try to get better so you can improve to be drafted higher despite the risk of injury. Now RB may be a bit of a different story because of the undervalue. The players I want back the most are as mentioned Beckwith and Pocic. We need Beckwith, he is the only decent returning LB. Even with him the core will be very young and unproven. I thought Pocic was by far our best, most consistent OL. His versatility is also great to have. I'd love White back but I think he should leave. Plus with Jackson and Toliver showing so much potential we should be fine there. Also LSU is bringing in an excellent group of CB's in the '16 class.
Truth is, they wouldn't have gotten where they are without both thinking they are the exception and actually being the exception.