What is the reason the Saints FO didn't restructure Taysom Hill's contract during the season?

You can't restructure a contract that is about to expire, there has to be additional years on the contract for the money to be shifted around to.

I believe what you're looking for is why didn't we EXTEND Taysom Hill during the season, the answer to which is what SaintRob said above: no need to extend him early when we will have first right of refusal AND potentially heavy compensation tied to him if he leaves. We hold all the cards with Taysom.