subnet_extend_of_vlan

You would then change the subnet mask used while configuring vlan. I would say it looks like

int vlan20
ip address 10.12.13.1 255.255.254.0
no shut

Thats fine.
My question is that apart from this where i have to make the change.

for example as earlier the Vlan10 ip address and subnet is 10.12.13.1 255.255.255.0
and this is the gateway for my LAN.

So on if there is any server in that subnet will be having the Gateway IP as 10.12.13.1 and the mask information will be 255.255.255.0

anyhow it will work even if i will not change its subnet from 255.255.255.0 to 255.255.254.0 but it will work from the local LAN only. From outside where ever the routing was done it was learning as /24 so i need to get the routing change done as well.

so that from wherever it was getting learned as /24 will learn as /23 and to make it fullproof and make sure that it will work fine. i think it is a good practice to change the sunet as well on all host where to /23

Please correct if i am wrong.

Depends if you are using static routing or OSPF/RIP etc.  You just need to change the IP address/subnet of the interface the gateway address is assigned on (10.12.13.1).  If you do this and not using static routing then the new subnet will be learned and entered in the routing table.  If using static routes you need to look on your switches/routers for these route definitions.

Don't forget if your using DHCP you need to change the scope/subnet.  DNS reserve lookup zones and if using AD, Sites subnets will need updating too.

Do i not need to change the subnet mask information from /24 to /23 on each host where ever the static IP was configured on the host/servers.

Thanks

Yes, if you are not using DHCP and only static I addresses, the subnet mask should be changed on the hosts as well