The Hunter brand The description of the spell is:
You select a creature that you can see within range, and mystically mark it as your quarry. Until the end of the spell, if you hit it with a weapon attack, you will inflict additional damage to the target 1W6, and you will benefit from any verification of wisdom (perception) or wisdom (survival) you make to find it , If the target drops to 0 hit points before the end of this spell, you can use a bonus action on a later turn to mark a new creature.
Specifically, the quoted sentence "a subsequent round of yours" means "a subsequent round after the one you initially cast the spell on". If the creature you have tagged dies, you can target a new creature each turn, as long as you concentrate on casting the spell.
The reason why "in a next round of yours" is to make it clear that you can not cast the spell to kill a creature with your weapon attack as an action and then move it to a new target in the same turn.
However, this is pretty much superfluous as you will only have one bonus action per turn anyway. Once you cast the spell, you will no longer have a bonus action that will allow you to move it to a new target until you still get another round. I suppose the designers just wanted to make it clear that the bonus action you use to cast the spell can not be the same as moving the mark to a new creature.