I think there is a small bug. Please look at the picture below. Since bounty period ended 23 hours ago, its grace period should not be ended yet. But it says it did.
In my opinion, this is a bug, and it has nothing to do with when the hourly job runs.
As you can see, the bounty was added at 16:25, but the grace period expired at 15:25.
I haven't been awarded the bounty yet and I don't know when it will happen, but the grace period either hasn't ended yet or it was only 23 hours long.
The bounty auto-award job does indeed run once an hour, there may be a gap between when the bounty ends and when that runs that you'll see an oddity like this, it's normal.