If you can ssh to the intermediate computer then you should be able to ssh further into the other computers (think of it as steps). If you don't want to do anything on the intermediate computer (only use it for getting through the firewall) then you can skip the creation of a shell on in by using the
-t parameter of ssh:
ssh firstname.lastname@example.org -t ssh email@example.com
This command will ask for the two passwords (if using password based authentication) on the two computers each time you are establishing the connection.
If you have to log in to the other computers more frequently then you could establish local port forwarding:
ssh firstname.lastname@example.org -N -L 3333:192.168.1.1:22
This establishes a port forwarding from your local laptop port 3333 to port 22 on the computer on the local network at 192.168.1.1. You have to use the ip address because you don't have access to the local DNS.
In a new terminal window you can connect now to the computer behind the firewall:
ssh user@localhost -p 3333
You can use this approach to establish local port forwarding to each of the firewalled computers. Be sure to use different local ports for each of them (e.g. 3334, 3335).