I definitely think there should be something more. In European football you have the league, the league cup and either the Europa League or Champions League to play for. Then on top of that there are smaller cups/matches like community shield, UEFA Super Cup etc. This is a way better format than just playing 82 regular season games then having playoffs. Having only 1 out of 32 teams being able to win each season is honestly pretty terrible.
Like sure, that makes the Stanley Cup really prestigious no doubt but whose to say there can't be other, less prestigious tournaments/cups to win as well? I think some type of cup that is being played out throughout the season (similar to Champions League) could be very entertaining. Hell, you could have it being played only through the first half of the season so the second half of it can be focused on making the playoffs (and winning the Cup) if you're worried about it interfering with the Cup hunt.
Alternatively, you could remove the AHL playoffs and instead have the top AHL teams go at it with the NHL teams that didn't make the playoffs. Presumably this wouldn't be too well-liked by the players on the NHL teams who failed to make the playoffs however so I don't know how you'd make it work. It would only work if there was relegation and they actually had something to play for.
I don't know, I definitely think there needs to be something more to NHL hockey than just 82 regular season games and the playoffs. Make the regular season shorter and add another tournament/cup. For the 31 unlucky fanbases it would be nice to at least win something in a season. Also it would be even more prestigious if the team who won the Cup was able to win the double. I really don't see a downside to this.