If you are instructed to cast a spell with overload "without paying its mana cost," you can't choose to pay its overload cost instead.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying (such as an overload cost), add any cost increases, then apply any cost reductions. The mana value of the spell remains unchanged, no matter what the total cost to cast it was.

Because a spell with overload doesn't target when its overload cost is paid, it may affect permanents with hexproof or with protection from the appropriate color.

Note that if the spell with overload is dealing damage, protection from that spell's color will still prevent that damage.

If you don't pay the overload cost of a spell with overload, that spell will have a single target. If you pay the overload cost, the spell won't have any targets.