I haven't been following the NHL rumors, so I don't know what they're considering, but I'm more hopeful they'll finish this season, then start the next one anywhere near on time.
They could potentially finish out the season by doing an altered version of the playoffs, perhaps with less games, to ensure they don't play a season without awarding the Cup. That's very important to the league.
The start of next season though will coincide with flu season which starts in the fall, and peaks around Dec-Feb. They won't know for sure how bad next winters wave of COVID-19 will be, until they start to get data on it, and the season will start before they get that.
My guess is they finish out this season by going straight to the playoffs in the summer. Then they'll start another abbreviated season in Feb-June after they see how the virus is spreading again. Perhaps just another straight playoff elimination if they have to wait too long.