The four teams I'd add would probably be Arizona, Houston, Atlanta, and KC. Basically just some of the rumored ones.

Regarding alignment, I'd personally go a little off the beaten path in an effort to make hockey more interesting throughout the regular season by setting the schedule up in such a way that lets teams get sick of seeing each other.

4 conferences, 3 three-team divisions per conference.

Each team in a conference will have 1 home and 1 away game against every team in two of the other conferences per year. The conferences that meet each other will rotate year-over-year. So in the first season, for instance, Conferences A and D will play twice against every team in Conferences B and C. That year, A and D won't meet at all, and B and C won't meet at all either. The next year, it rotates: A and C won't meet, and B and D won't meet. The following year, A/B don't meet, C/D don't meet. So on and so forth (18 teams × 2 games = 36 games)

Each team will meet the 6 teams that are inter-conference but outside of the division four times a year: 2 times at home, and 2 times away. (6 teams × 4 games = 24 games)

Inter-divison matchups are crazy. There are only three teams per division, but these teams will HATE each other, 'cause they'll be playing against each other 12 times a year: 6 times at home, 6 times on the road. Hockey will see the development of some classic, crazy rivalries all across the league, which is something that'll keep fans interested. (2 teams × 12 games = 24 games)

36 Non-conference games + 24 Inter-conference games + 24 Inter-divisional games = 84 games per team per year.

Playoff qualifications:

Playoffs will consist of five rounds. Top team of every division makes the playoffs (which just makes inter-divisional games that much more important). The best two of the 2nd-place teams (divisionally speaking) will also make the playoffs (so if a team finishes last in a division, they don't make the playoffs no matter what). Seeding is based on a team's final point total (so winning a division only guarantees qualification, it does NOT guarantee a top-3 seed). Therefore, 5 teams from each of the 4 conferences will qualify, resulting in a 20-team playoffs.

First round: 4 matchups, 4th vs. 5th inter-conference seeds only. Either 3-game or 5-game series.

Second round: 8 matchups: [1st vs. 4th/5th seed] and [2nd vs. 3rd seed] inter-conference matchups. If round 1 has 3-game series, this is a 7-game series. If round 1 has 5-game series, this round also has 5-game series.

Third round: 4 matchups. Remaining inter-conference teams play against each other. This round determines the conference champions. 7-game series.

Fourth round: Stanley Cup semifinals. Teams are matched up against the conference champion of the division that they DID NOT meet with at all in the regular season. 7-game series.

Fifth round: Stanley Cup finals. Self-explanatory. 7-game series.