Yes, if I statically configure a client on the new subnet (192.168.12.0) it can ping stuff in the old subnet and get to the Internet.

So, for example, if I have a VPN road warrior set up, and I'm somewhere where the network is 10.0.0.1/24 but the VPN tunnel network happens to be 10.0.0.1/30? Changing the client network address ...