Here's another idea: when loading a game, add the option to adjust the timer with the mutual consent of the players.

Just in case, here's why this might be useful: when both players are running out of time, they might want to add more time and continue the game comfortably. This can be implemented in another way - by adding a button in the game that says "add time," and when clicked, a window pops up for the opponent asking, "Do you agree to add time?". If they agree, the time is added to their turn. (By default, say, +30 minutes)

P.S. Another global benefit of this feature is that players will be more willing to agree to reduce the timer in tournaments.