It's not clear if you are just wanting to connect to an additional machine on the same network as Linux Machines 1/2 or if you need to jump to another network beyond them (for example Machine 4 on 172.22.1.x).

If you're simply trying to get to another machine on the same 192.168.1.x network as Linux Machine 1/2, use an actual SSH Tunnel (found under Putty's Connection -> SSH -> Tunnels menu). You can set up multiple tunnels to as many machines as you need to.

If you are actually trying to get to a third network, you can open up another instance of Putty that connects via a tunnel defined on your original Putty instance.

I've only ever done this with 2 hops so I don't know how it will perform. It's not an ideal solution, but it may work if this is your only option.

Edit: If you're not limited to Windows/Putty, this (Transparent Multi-hop SSH) may do what you want. If you are limited to Windows, Cygwin may have some more flexible SSH tools.