Six divisons
If we must keep six divisions, the easiest change is to move Winnipeg and Toronto to the West and move Dallas and Nashville to the South, which looks like this:
Atlantic: PHI, PIT, NYR, NJD, WAS
Northeast: NYI, BOS, MTL, OTT, BUF
South: DAL, NSH, TB, FLA, CAR
Central: TOR, DET, CHI, CLB, STL
Northwest: CALGARY, EDM, COL, WPG, MIN
Pacific: VAN, ANH, LA, SJ, PHX
If the Islanders then move to Kansas, bring Columbus East and you get
Atlantic: PHI, PIT, CLB, NJD, WAS
Northeast: NYR, BOS, MTL, OTT, BUF
South: DAL, NSH, TB, FLA, CAR
Central: TOR, DET, CHI, KANSAS Islanders, STL
Northwest: CALGARY, EDM, COL, WPG, MIN
Pacific: VAN, ANH, LA, SJ, PHX
I think that a three conference system would work well.
Here are my three conferences (feel free to suggest others): Northeast, West, and South. Each has 10 teams:
South
Carolina, Florida, Tampa Bay, Nashville, Dallas, St. Louis, Columbus, Colorado, Minnesota, and Phoenix
Northeast
Montreal, Ottawa, Buffalo, Boston, NY, NYI, NJ, Washington, Pittsburgh, and Philadelphia
Northwest
Toronto, Detroit, Chicago, Calgary, Edmonton, Vancouver, San Jose, Anaheim, Los Angeles, Winnepeg
Playoff system:
Two possibilities: each conference winner ranked 1 2 3, followed by the remaining teams from any conference to 16.
Top 2, 3, or 4 teams in each conference make the playoffs. As long as there are matchups within a conference then teams match up within conference.
Examples:
If in round one, the three conferences provide 4 4 and 8 teams, round 2 they provide 2 2 and 4 teams in round three they provide 1 1 and 2 teams, and then the conference final is the winner of the deepest conference vs. the winner of the other playoff series.
In gets more complex if the three conferences provide 5 5 and 6 teams. The fifth ranked teams of two conferences play each other. In round 2, the conferences provide 3 2 and 3 teams. Again, the third ranked teams from two conferences play each other. In round 3, the conferences provide 2 1 and 1 teams, so then we're back at the situation above?).
Note that you could argue that giving the last placed team an opponent who is another last placed team benefits those last place teams. So you might, instead, pit the middle teams against each other. In the case of two conferences providing 5 teams each, each conference's third ranked team would play off against each other. In the case of two conferences with 3 teams each, the second ranked teams would play off against each other.
Change the number of games played in a season
I would advocate 76 games: each team plays 4 games against the 9 other teams in conference, and 2 games against each of the other 20 teams -- but if each team plays each conference team 4x plus every team in one other conference 2x plus every team in the third conference 1x you end up with 66 games a year, which would be better for our frequently injured players.