Well if port 443 is used by the webserver you cannot reuse it for something else (actually there are ways to do such things) What I would suggest is to have the server create an outbound SSH connection to another box, quite likely that the firewall policy will allow this.
Do something like this:
ssh -L 2222:localhost:22 some.other.server
(Perhaps even make it persistent using a service. https://gist.github.com/drmalex07/c0f9304deea566842490)
Then login to some.other.server and do this:
ssh -p 2222 webserveruser@localhost
Or let your automation connect to this port.
To take it one step further you could use the jump option to hop directly through the intermediate server. Your .ssh/config could look something like this.
Host intermediate.server HostName intermediate.server User someuser IdentityFile ~/.ssh/id_rsa Host webserver HostName localhost User someuser Port 2222 IdentityFile ~/.ssh/id_rsa ProxyJump intermediate.server
effectively allowing you to do this:
ssh webserver
Gotta love ssh!