Are you sure that they are blocking SSH packets, or just port 22? If they are blocking the port, it's good practice to use something other than 22 anyways
I keep a machine which has sshd listening on the IMAPS port (993) for when I'm traveling. It's amazing how many free networks don't allow ssh, but with -J and sshd on port 993, that really doesn't matter.
Agreed. You can host both SSH and HTTPS on port 443. I know this used to be possible with HAProxy, but now Nginx can do it as well. This way you are hosting normal HTTPS traffic when a browser is used and SSH otherwise.
Now, if your company is actually blocking the SSH protocol, you’ll have to do something like tunneling SSH through SSL, which is also possible… but not as easier IIRC.