Question regarding adding collision and comprehensive coverage shortly before renewal deadline…

Background

I currently drive an old beater, so I have not bothered with comprehensive and collision coverage. I am about to get a new car, and would like to do a 24-hour test drive through CarMax before purchasing. Unfortunately, I need comp/col coverage to do so.

Question

My current cycle ends 8/23 and I plan on switching providers for my new car. If I were to increase my current coverage to include comp/col, and then not renew my policy come the 23rd, would I only have to pay the additional cost for these next two weeks?

Hopefully that makes sense. Thanks!