No. You have to extend him as part of - not prior to - the restructure. In general restructures just shift some of a player's current year cap charges to future years. The typical way to do this is to convert a portion of the player's salary and/or roster and workout bonuses to a signing bonus. The player is happy because he gets his money a little earlier than he would have. The team is happy because they delay their cap hit.
Clay is a good example. He has a $10M roster bonus due in the offseason. That will all count toward the 2016 cap if left in place. But it won't be. It was just done that way to keep Miami from matching his deal. They will restructure it to a roster bonus, which is amortized over the rest of the contract (up to 5 years or the end of the CBA). He has 4 years after this one on his deal so the cap hits will be $2.5M in each year from 2016-2019 instead of $10M in 2016. That clears $7.5M in 2016 and adds $2.5M to seasons 2017-2019.
To free up any worthwhile amount of cap space for any significant amount of time from Mario's contract he'd have to have some years added. How long he is expected to be a highly productive is of paramount importance because if he is released, retires, or is traded any unaccounted for signing bonus dollars get accelerated into the current year (or the following year if after June 15th).
This offseason Mario and the Bills could, for example, agree to a 2 year extension (through 2019) and convert his whole $2.5M roster bonus and $9.5M of his salary ($12M total) to a signing bonus. That would spread out the $12M cap hit in 2016 to four $3M cap hits in each season from 2016-2019. His 2016 cap hit would be reduced by $9M, but his cap hit in 2017 would increase by $3M and hed also have additional $3M cap hits in 2018 and 2019.
In short, there's no free lunch but you can defer the cap hit. An argument for deferring cap hits is that the cap almost always rises so taking a hit in future years will take up a proportionately lower percentage of the overall space. The big arguments against it are that sometimes players will lack motivation after they get one big payday and then a relatively small salary especially if the restructure makes it difficult for the team to cut or trade them. That's the second argument against it - it can really create issues if a player becomes a big off the field problem, gets suspended or retires unexpectedly due to injury. New Orleans is going through this now.