If you configure the new router with an address of 192.168.1.5, disable its DHCP server and connect one of its LAN ports to a LAN port on the old router, it will perform only as a basic Network Switch or Wireless Access Point. It *WON'T* perform as a Router to serve internet access to the connected client devices - they'll continue to get internet access from the old router, albeit possibly via a connection to the new router.
I cannot see the merit in not simply connecting all devices to the new router for both LAN and Internet access, particularly if everything operates in the 192.168.1.0 subnet. If the routers were to operate in different subnets, and the computers had two network adapters installed, they could connect simultaneously to both routers. You could then selectively disconnect one router from the internet (phone line).