Duane,
I think that should work without any problems. A more practical and generally used approach is using " secondry ip addresses " under the interface.
My suggestion for you would be to define " 172.20.6.0 255.255.252.0 " secondry ip address range under the interface of the router which is currently acting as the gateway to your Lan. Onec that is done then try migrating your users to the new segment IP and once your users are on the new ip address range, you can take out the older ip address under the router interface.
HTH,Please rate if it does.
-amit singh