If the 192.168.1.x range is using a 255.255.255.0 subnet mask. Then you could simply use this same mask for your 192.168.3.x range instead of the 255.255.252.0 mask. This way the two subnets are disjoint and you will not cross the 192.168.1.x range.
If you have a layer 3 switch you can most likely create a L3 Vlan to route between the two. If not use a router or Linux machine as a gateway and configure routing.
If you have implemented DHCP then you could choose a different IP range like 10.0.0.x 255.255.0.0, depending on how many Hosts Networks you need adjust the subnet accordingly, and allow DHCP to asign the PC's automatially, all you have to do is statically set any servers if thats how you cofig your servers and change Router/Firewall config.
how you do it is up to you, you need to plan first, make sure you have thought of everything!!
To implement from what you said, its sounds like it wont be complicated, unless you have a specific IT policy or IP restrictions internally.
Let me add some more info here, while we started with a 192.168.3.x with a subnet of 255.255.255.0 we ran out of hosts.
We expanded the range with a 255.255.252.0 giving us 254 x 4 hosts, the range of hosts cover the 192.168.0.1 – 192.168.3.254 block of address.
The problem is, the 192.168.0.1 to 192.168.1.254 block is used by many home routers.
The options, as I see them...
1> Retain the current IP scheme and just open the subnet up for more hosts and exclude the 192.168.0.0 – 192.168.1.254 range.
If one subnets to 255.255.240.0 – This extends the useable range out from 192.168.0.1 – 192.168.15.254.
This makes for 4094 hosts before excluding the first 500+ hosts to avoid the 0.1 to 1.254 problem.
2> Deploy a new IP scheme on a logically different network outside the private address range. Setup multiple routers to handle the bandwidth load of 250 workstations accessing 20 servers, so the migration can be completed in stages.