Not sure if it has to go that far. Probably it's just blocking port 22.

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.