You can Leave the Spell Active
Under the rules for spells with long durations there's quite a bit about those which last until the next time you make your daily preparations:
If a spell’s duration says it lasts until your next daily preparations, on the next day you can refrain from preparing a new spell in that spell’s slot. (If you are a spontaneous caster, you can instead expend a spell slot during your preparations.) Doing so extends the spell’s duration until your next daily preparations. This effectively Sustains the Spell over a long period of time. If you prepare a new spell in the slot (or don’t expend a spell slot), the spell ends. You can’t do this if the spell didn’t come from one of your spell slots. If you are dead or otherwise incapacitated at the 24-hour mark after the time you Cast the Spell or the last time you extended its duration, the spell ends.
You can do this as long as you're alive and well, so taking a six week journey to save someone could work.
To the broader question of not making daily preparations at all, you could avoid getting proper rest:
After you rest, you make your daily preparations, which takes around 1 hour. You can prepare only if you’ve rested, and only once per day.
Otherwise the language suggests that making daily preparations happens automatically after resting, but as a discrete activity.
Deciding not to is something your character would reasonably have choice over, but otherwise staying awake would work to avoid making daily preparations.