At the end of the tournament any spare/leftover points get added automatically to your archive storage up to the limit which is determined by the perks level.
If you are short of points to get a chest and you have enough in storage then the archmage or a mage can use them to complete the chest.
Only enough to complete the chest get used, the balance stay in storage.
My opinion: depending on how many chests you get, it's worth having the storage level at high enough to get the next chest.