Yes, you can.
The hunter's mark spell description says:
You choose a creature you can see within range and mystically mark it as your quarry. Until the spell ends, you deal an extra 1d6 damage to the target whenever you hit it with a weapon attack, and you have advantage on any Wisdom (Perception) or Wisdom (Survival) check you make to find it. If the target drops to 0 hit points before this spell ends, you can use a bonus action on a subsequent turn of yours to mark a new creature.
Specifically, the quoted sentence uses "a subsequent turn of yours" to mean "a subsequent turn after the one on which you initially cast the spell". If the creature you've marked dies, then on any subsequent turn as long as you're maintaining concentration on that casting of the spell, you can mark a new creature.
The reason it says "on a subsequent turn of yours" is to make it clear that you can't cast the spell, kill a creature with your weapon attack as your action, and then move it to a new target all on the same turn.
That said, this is pretty redundant because you only have one bonus action per turn anyway; once you've cast the spell, you no longer have a bonus action with which to move it to a new target until you get another turn anyway. I suppose the designers just wanted to make it clear that the bonus action with which you cast the spell can't be the same one with which you move the mark to a new creature.