You can address from outside toward center. This will cause lost of connectivity to readdressed devices until complete path has been readdressed. Or, you might use secondary addressing, on routers, to support both old and new addresses until other devices are on the new addresses. When you change router's primary address, there's often a momentary service interruption, although if done with a clip and paste, some hosts might not notice.

e.g.

(now)

interface Ethernet 0

ip address x.x.x.x 255.255.255.0

(during remote device transition)

interface Ethernet 0

ip address y.y.y.y 255.255.255.0

ip address x.x.x.x 255.255.255.0 secondary

(after remote device transition)

interface Ethernet 0

ip address y.y.y.y 255.255.255.0