Sounds like a possible leaking rear main seal which is dripping oil on the flywheel overnight, then after the first use it gets burned off.
The problem though is that over time it'll get worse as the oil begins to contaminate the friction material on the clutch disc.
Check under the car at the vent hole between the tranny and the motor - my suspiscions of a leaking seal will manifest itself as a tell tale drip of oil coming out of this vent.
