How to automatize a ssh tunnel over various hosts?

One way to do this is with a proxy auto-config (PAC) file and a SOCKS proxy. You would then set up the forwarding in your ~/.ssh/config file to forward all requests on some local port through a SOCKS tunnel to the remote machine, and have the PAC file route requests matching the host name through that tunnel.

For example, a PAC file with content:

function FindProxyForURL(url, host) {

  alert("url" + url);

  if (shExpMatch(url,"*myhostname*")) {
    alert("Using special tunnel");
    return "SOCKS5 127.0.0.1:9999";
  }

  // All other requests go directly to the WWW:
  return "DIRECT";

}

coupled with an entry in your ~/.ssh/config file such as:

Host PH-*
     IdentityFile ~/.ssh/id_rsa  
     User username  
     ProxyJump PC-SSH
     DynamicForward 9999

when you execute ssh -Nf PH-05 would open local port 9999 and forward all local traffic on that port through your ssh tunnel to PH-05. The options -Nf puts the process into the background, but the config file will open this forward for any ssh connection to PH-*, so you may prefer to create a specific alias for just the tunnel, e.g., PH-05-tunnel, and only have the DynamicForward included there. Alternatively, if PH-SSH can access PH-05:4040, then you could just create a PH-SSH-tunnel rule, and set the hostname to PH-SSH, eliminating the ProxyJump part.

Then, depending on your setup (firefox has proxy settings in the browser, chrome/chromium use system proxy settings) you tell your proxy manager to use the PAC file and then you can point your browser to, e.g., PH-05:4040 and your browser should render this page.