Tectonic Giant doesn't change when you can play the exiled card. For example, if you exile a sorcery card, you can cast it only during your main phase when the stack is empty. If you exile a land card, you can play it only during your main phase and only if you have an available land play remaining.

Playing an exiled card causes it to leave exile. You can't play it multiple times.

An ability that triggers when a creature becomes the target of a spell resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

If you don't play the exiled card that you chose, it remains in exile with the card you didn't choose.

Players can cast spells and activate abilities after the triggered ability resolves but before the spell that caused it to trigger does.

If a spell targets Tectonic Giant more than once, its ability triggers only once.

You choose which of the exiled cards you'll be able to play while the ability is resolving. You can't wait to pick one later. The other card remains in exile.