Okay... This is very possible with Putty though not as easy as with OpenSSH on a Linux machine. I would very much recommend setting up an SSH identity key and installing it on the boxes that you are connecting to and use the Putty Agent (pagent.exe) key forwarding agent. That said here is the steps to take.
First, create a Putty connection profile for your work Linux box that you can reach from home. You'll want to be sure under
Connection -> Data that you set your
Auto-login username. Then under
Connection -> SSH -> Auth be sure that
Allow agent forwarding is checked. Now save this profile and make note of the name you call it (ie- work-jumpbox).
With this setup you now want to setup a second profile for the box behind the work jumpbox. Set this profile up with the correct host name/IP address and port of the webserver in the office. You then want to go to
Connection -> Data -> Proxy and set the
Proxy Type to
local and set the
Telnet command or local-proxy-command to
plink -load work-jumpbox -nc %host:%port\n (replace work-jumpbox with whatever you called your previous profile for the work linux box.
Now save this profile (ie- work-webserver) and when you want to connect to it load it and click connected it should log you into the office web server after it logs into the jumpbox silently.