With the flat cap for the next 3 years, I just don't see how they're going to give Hyman much more than what guys like Kapanen/Kerfoot/Johnsson earn right now.
That being said, we are in a new economic reality. "Accessory Players"; which Hyman is a stereotypical example of are likely going to face the brunt of the hit from a flat cap, older veterans tied into long term deals, and young players who's teams are afraid to lose them. I think the days of the Justin Abdelkader's of the world getting $4m plus long term are gone.
That being said, as others have mentioned, I wouldn't be surprised to see Hyman play the long game. He's a UFA, so he can get trade protection. The Leafs are probably one of the only teams in the league that can afford to front-load and bonus-laden a deal right now. The other top teams around the league are all going to be quite cap-strapped next summer anyways. If his alternative is going somewhere where he knows he's unlikely to be successful, or taking a major haircut to remain in Toronto -- there's a lot about his personal background that would lean him towards taking that haircut.