As someone who worked in hotels for many years, I always gently ask people to let someone at the hotel know when you leave. The reason is that the hotel can't clean a room if they don't know it's vacant. And without clean rooms, hotels can't allow guests to check in early.
You don't need to visit the Front Desk. Just pick up the phone, dial "0", and tell whoever answers that you're leaving. In any decent hotel, that person will pass along your room number to the right folks. It shouldn't take more than 60 seconds. If no one answers right away, then just leave knowing you tried.
The simple act of letting someone know you left might mean that the next guest staying in that room can get in earlier than normal check-in time. For me, that's worth taking 60 seconds out of my day to do.