The only thing the NHL can do to minimize the tax advantage between the teams in U.S. states with no state income taxes and the teams in states/provinces with state/provincial income taxes is to prohibit signing bonuses. It's the signing bonuses which can make a difference for a player in the end.
Steven Stamkos' contract is a prime example of it:
2016-17: $8.5m (SB) + $1m (salary)
2017-18: $8.5m (SB) + $1m (salary)
2018-19: $8.5m (SB) + $1m (salary)
2019-20: $8.5m (SB) + $1m (salary)
2020-21: $8.5m (SB) + $1m (salary)
2021-22: $6.5m (SB) + $1m (salary)
2022-23: $5.5m (SB) + $1m (salary)
2023-24: $5.5m (SB) + $1m (salary)
Totals: $60m (SB) + $8m (salary)
Signing bonuses are usually paid during the off-season and taxed in the state/province the player are resident in. It's the salary, which are paid out 13 times during the regular season, are taxed in the state/province the player (and the team) are when the salary is paid out.
If the NHL had prohibited signing bonuses then Stamkos would not been so eager to give any major discount to Tampa because much more of his salary would be taxed. He would then demand another $1-3 million to the annual salary on his current contract. That would lead to that Tampa could not have the roster they've now and probably not been able to achieve a potential three-peat because the other star players would also demand higher salaries which mean less competent support players on the roster.