Like Toe Nash, my mind also went to European soccer, where a team can be playing in their domestic league, as well as a domestic cup tournament, as well as an international tournament like the Champions or Europa Leagues. Combining trying to win any/all these competitions with trying to qualify for the international leagues for the following season and trying to avoid relegation, there are a lot of things to play for, and seasons can stay interesting and feel successful in multiple ways.
I'll also say that system, while I think it's great, does have the issue that there are a lot of games, and it seems to increase as leagues look for more ways to play games and more ways to make money. Combine that with players constantly going off to play with their national teams in addition to the club games, and injuries seem to be becoming more and more of a problem as players get overworked. This is something soccer needs to address, but I still really like the basic structure.
But anyway, I have long felt it would be better for baseball if there were other ways to have successful seasons besides just being the 1 of the 30 that wins the postseason — which, as we're discussing here, has become more and more of a tournament independent of the regular season. What seems simplest to me is to just give "the AL pennant" and "the NL pennant" to the team in each league that finishes the regular season with the best record, and have there be big trophies for that and maybe some other reward, like automatic entry into the postseason the next year. Then keep the World Series as-is as the culmination of the post season tournament. Often the pennant winners would be different than the WS winner, and that's fine. Sometimes a pennant winner would also win the WS, and it would be a particularly special season. I can think of other formats, but this one is the least departure from the way we play now.