This isn't really news; we knew since the beginning of the playoff run (really, earlier than that) that the Lightning and Stamkos hadn't had any discussions at all about a new deal during the season. Both sides were focusing on the task at hand -- winning a Stanley Cup, something they came very close to doing.
Following the defeat at the hands of the Chicago Blackhawks in 6 games in the Final, Stamkos' next move wasn't "go talk to my agent and start working on a new deal".
Understandably, it was "take some time off to decompress". He went on a short vacation. He took some batting practice with the Tampa Bay Rays and a few of his teammates. You know, stuff that isn't hockey.
The July 1 date, when he was officially allowed to begin re-negotiations and re-sign with the Lightning, has come and gone. It's been more than a week since that date. Still, there's absolutely zero reason for panic; this is the beginning of a process, one that might take a while. Consider the last time Stamkos was up for a new contract:
Lightning general manager Steve Yzerman announced the deal, saying that even though the deal took several months to come together that there never was a doubt that Stamkos -- a restricted free agent -- would remain with the team.
"I never even thought of looking anywhere else," the 21-year-old Stamkos said, adding that contract negotiations were a learning experience.
The process took "months", which means it began before July 1, and still didn't get done until nearly August. (Stammergeddon, it was called at the time.)
This isn't going to happen overnight. His last deal -- after a similarly long playoff run in 2010-11 -- was signed on July 19. He was an RFA then, however, with less options available to him. As a pending UFA, there's more to weigh. This decision might take even longer. That's okay.
Comparisons to the timeline on Patrick Kane and Jonathan Toews' extensions are inevitable, but those two players had completely different situations. Stamkos has his own decision to make. The conversations with his friends, family, and his agent about his future are happening now, and we're still many weeks away from training camp back in Tampa.
Stammergeddon 2.0 has begun in earnest, particularly with some of the larger (ahem) "Northern" media outlets. Remember though, this is priority number one for Steve Yzerman. All of the tea leaves we've been able to read so far suggest staying in Tampa is the way Stamkos has been leaning.
Stamkos will talk to his people. His agent, and Yzerman, will meet.
Then we'll see where we stand.