Targeting Hex over long durations

Previous questions confirmed that Hex can last through multiple encounters (assuming my warlock’s concentration doesn’t get broken by a club to the head or the like) and at higher levels even through a short rest.

My question is how targeting Hex works when trying to keep it up for extended periods of time.

Can I just cast it on nothing and keep it up until a valid target presents itself or is an initial valid target required?

Then later, does changing the target of the Hex require that the original target is brought to 0 HP?

Once the initial target dies does the spell stay active even if I don’t immediately choose a new target?

My suspicion from reading the spell is that it needs an initial target, but then from that point on I can just keep it up after that target dies until it either expires or a new target presents itself. Is this correct?