You can configure your local tmux to have a key press send the prefix command to a remote tmux with a single keypress.

For example, to bind C-a to control the remote session:

   bind-key -n C-a send-prefix

Binding isn't an option as only defaults are available on target machines (can't save .screenrc file). There can be thousands of target machines so binding doesn't scale.

I'm sorry I wasn't clear. This would be a binding in your local tmux configuration that sends the prefix command to whatever is in your current pane, allowing you to have a key act as a 'remote prefix' button.

I agree that bindings still don't scale incredibly well depending on the number of host machines you need to ssh from. I think your screen workaround is a pretty neat solution.

I suppose you mean that you run tmux locally and then you ssh into a server and run screen remotely?

If so, I wonder how hard it would be to make tmux into a client of itself using control mode. I only learned about control mode when I discovered iTerm2's tmux integration, whereby remote tmux windows materialize as "actual" terminal windows on my desktop. However, for the same reason that iTerm2 can resize a window, create a new one, show and hide them, etc, I would _imagine_ it's possible to have your remote tmux windows materialize as tmux windows in your local tmux, and thus would be managed using only a single prefix key.

I would guess such a capability would need to be baked into tmux, but I actually wonder how much code would actually be required, since control mode already exists and clearly works quite well.

ssh to remote server over VPN then tmux attach, then connect to target machine and launch screen to work multiple sessions on target machine as it's not networked.