TL;DR - You can pre-cast hex prior to encountering an enemy by targeting a harmless bug (a "creature you can see") and crushing it later using your free item interaction.
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?
As long as you can maintain concentration, yes. Previous answers did a good job of explaining this.
However, it is still possible to cast hex in advance in certain scenarios. The hex spell requires that you target a creature you can see within range, not that the creature be a monster from the Monster Manual. As such, you could technically cast your spell on an insect, which you keep on your person in an accessible fashion. For example, you could wear a small cage or canvas pouch on a cord around your neck, or maybe tied to your belt.
A single trapped insect shouldn't require an attack roll to kill. You grab the pouch ans squeeze, which should be well within the range of actions that can be performed with your free item interaction (see PHB, p.190). In fact, some of the actions listed as permitted seem more difficult in nature:
- Fishing a few coins from a pouch.
- Grabbing a potion from your backpack.
- Removing a ring from your finger.
And so on. All of these require you to grab something and then perform the action. Killing the insect in the pouch only requires you to grab the pouch, so it should definitely be possible. Your GM might disagree, however.
If your GM does agree, this means that you can catch a grasshopper or something, hex it, then put it in a small pouch that you carry. You then take a short rest to get your spell back, and you can go adventuring. Once you encounter baddies, assuming you are still concentrating on the spell, you can use your item interaction on your turn to grab the pouch real hard. The bug dies, allowing you to use a bonus action to target one of the baddies with your hex. You then use your action as you see fit.
In most natural environments, finding an insect or other suitable critter should be pretty easy, perhaps requiring a DC10 or lower survival check (you're not trying to find enough insects to feed yourself for a day, only a single one to use as a "spell component"). This has the added benefit of looking very warlock-like, too, crushing critters to curse your enemies.