What age and type of property do you have? 60 litres is enough for a small bath, or medium shower. I assume you'd have mentioned if you have an automatic watering system for a garden or greenhouse. It's not unknown for a neighbour's extension or garden tap to get connected to the wrong service pipe.
If possible, monitor your meter to see whether the consumption is, e.g., a five minute burst at 12 litre / minute or a continuous trickle of 1 litre / minute for an hour. If you can see it happening, lift an inspection cover or two to try to figure out whether it's being discharged into a drain and, if so, which one. Listen carefully to see if you can hear water running somewhere.
First, I'd check carefully to make sure that no overflows are running. Relatively modern toilet cisterns have flap valves rather than siphons and overflows that discharge into the bowl. You have to look carefully to see if this is happening; dry the sides of the bowl and see if it gets wet without being flushed.
Next, what sort of hot water system do you have? Does the unexplained consumption happen during the period a cylinder is being heated? Is there a tundish that runs (unvented cylinder) or overflow from an expansion tank (vented cylinder)?
I'd also liaise with my neighbours and ask if they do something that uses roughly that amount of water at that time. Try turning off your water at your meter between 2000 and 1100hrs and let your neighbours know in advance that action affects them. Don't do this without warning your neighbours first; when their water supply is cut off without warning, people sometimes open taps fully and forget to close them causing a flood when the supply is restored!
Have a careful look around the property, inside and out, for any signs of leaking water. Wet masonry, a patch of grass that is growing faster than the rest, etc.
Having checked the 'obvious' issues, I'd report the problem to the water supplier who fitted the meter and ask them to investigate.