You cannot have overlapping networks. Network addressing for a network must be contiguous. If you use part of a parent network, then the parent network no longer exists, only the subnets you have created from the parent network.
If you try to use part of a network somewhere else as you have described, hosts on the original network cannot access the part you have moved. The way a host uses a gateway (router) is that all packets are encapsulated frames and delivered on the local network. A host will mask the destination address on a packet with its mask and compare that to its own masked address to see if they are on the same network. If they are, it will use ARP to get the destination MAC address, but the host on the remote network will never respond, so the packet will be dropped. If the destination IP address is on a different network, the host will use ARP to get the gateway address and encapsulate the packet in a frame destined for its gateway. The router will get it and look up how to reach the destination network. If the address is in the source network, which will be the network from where the packet originated.
Also, devices addressed in one network (
192.168.0.96/28) cannot connect to a different network (
192.168.10.0/24) unless the addresses are changed to the new network. For example, when a device addressed in the old network is attached to the new network, it cannot communicate with other devices on the new network, including the gateway of the new network, because it will need to send traffic to its configured gateway, which must be in the same network (old) as it, but the gateway is in the new network. It would require a gateway to reach the gateway, and you cannot do that.
You would need to create a completely separate new network, not add to an existing network.
What you propose cannot work.