Using an already established SSH channel
It's very simple with recent enough versions of OpenSSH if you plan in advance.
Open a master connection the first time. For subsequent connections, route slave connections through the existing master connection. In your ~/.ssh/config, set up connection sharing to happen automatically:
ControlMaster auto
ControlPath ~/.ssh/control:%h:%p:%r
If you start an ssh session to the same (user, port, machine) as an existing connection, the second session will be tunneled over the first. Establishing the second connection requires no new authentication and is very fast.
That's pretty easy to achieve using the nc tool and ssh tunnels.
1. Open ssh tunnel
In your ssh session, type ~C on a new line. You will get the ssh "service console" prompt which looks like this:
ssh>
Type in the local forward command to open an ssh tunnel:
ssh> -L22000:targethost:22001
Forwarding port.
Where targethost is the hostname or IP address of the machine you are connected to.
Now, assuming the ssh server on the target machine wasn't configured to forbid tunnels, you have the desired connection forwarding: ssh client on your machine listens to port 22000, and it will forward any traffic sent to it to the 22001 port on targethost.
2. Start a network server on the remote machine
This is as simple as entering into your already open ssh session the following command:
remote$ nc -l localhost 22001 | sh
This will start a TCP server listening on port 22001 – which is the target port of our ssh tunnel – and route the received data (presumably, shell commands) to a targethost shell instance.
3. Send your script over the tunnel
local$ cat yourscript.sh | nc localhost 22000
This will send the script's body to your ssh tunnel and will end up being executed in a shell on the targethost. You will see script's output in your terminal with ssh session.
I'll also note that ssh tunnel (step 1.) in this scenario isn't strictly required; you could as well start the server open and connect to it directly over the internet. However, you will need to use the tunnel if the target host can't be reached directly (e.g. is behind a NAT), or ssh encryption is desired.