SORNing a car I just bought

[BHML]

Bought a car this afternoon and got driveaway insurance, so it is now safely tucked up in my drive way, off road.

However, I won't need the car for a couple of months. Do I declare SORN immediately using the part of the logbook that the seller gave to me, or do I need to wait for DVLA to send me the full new logbook and then do the SORN? Is there any chance I'll get fined for not SORNing before the new logbook arrives?

Thanks!