Slayer’s Prey can activate multiple times in one turn, if those activations are against different creatures
First, I’ll just note that the scenario you’ve described works exactly as stated, you can target one creature with Slayer’s Prey, attack them on your next turn and then target a new creature with Slayer’s Prey and attack them on that same turn.
This is because bonus actions can be used in the middle of actions (or perhaps you’re using something other than Extra Attack, which would also work). And because you only have one bonus action per turn, you are correct that you would have to use Slayer’s Prey earlier, and wait until the next round to capitalize on this strategy.
Thus, we have Slayer’s Prey on a target, we take the Attack action, attacking them and dealing extra damage; we then use Slayer’s Prey to mark a new target, and use the other attack of the Attack action from Extra Attack (or some other similar series of events) and deal damage to another creature marked by Slayer’s Prey. You have already quoted the relevant part of the feature:
(…) The first time each turn that you hit that target with a weapon attack, it takes an extra 1d6 damage from the weapon. (…)
The restriction of it applying once per turn is also a per target restriction, meaning that having a new target gets around this restriction and allows it to activate more than once per turn. Though at most only twice as you have already used your one and only bonus action for that turn.