I'm no cap expert either but if I understand correctly what I read here: https://www.reddit.com/r/hawks/comments/6kh944/2017_nhl_free_agency_a_guide_for_the_chicago/
then the Hawks options are as follows:
1: Get cap compliant by the beginning of the season and put Hossa on LTIR after game 1. This would give full cap relief for his contract but the options become more limited when they try to use the cap space because most teams aren't looking to make trades that early in the season.
2: Put Hossa on LTIR prior to the start of the season. In this case, whatever the amount they were over the cap with Hossa is the amount of cap savings. The rest of his cap hit still goes towards the cap so for instance, if they were in the situation they have now with 77,135,628 cap hit, and they put Hossa on LTIR, then the cap savings would be 2,135,628 and the rest of his salary would count against the cap (3,139,372). If I understand that amount would count against the cap all year no matter what they do (unless they trade him).
2A: This is what I think they are doing. If they spend right up to the cap limit plus Hossa's cap hit, therefore 80,275,000 and then they put Hossa on LTIR, then his full salary will be cap savings and their cap spending will be at 75M. So what they need to do is maybe sign 1 or 2 guys (for instance Oesterle and Bouma) and then bury their contracts once the season starts to give some wiggle room for trades and deadline trades.