I assume you mean copying text with copy mode in tmux?
tmux can set the host clipboard using the OSC 52 escape sequence, but it needs to be aware the outside terminal supports this and the outside terminal needs to be configured to allow it.
With only one tmux, you probably have TERM=xterm outside tmux and a terminal that allows this by default, so it will work without any configuration changes.
However, with two tmux, the inner tmux will have TERM=screen or TERM=tmux and an outside terminal (the outer tmux) that does not allow this by default.
So to make it work you need to do two things:
1) Configure the outer tmux to allow the clipboard to be set by programs inside with set -g set-clipboard on (don't forget to restart tmux entirely after changing .tmux.conf, or do this also from the command prompt).
2) Configure the inner tmux to set the clipboard when the outside terminal is also tmux by adding an Ms entry to terminal-overrides (change screen* to tmux* if you have TERM=tmux inside tmux): set -as terminal-overrides ',screen*:Ms=\\E]52;%p1%s;%p2%s\\007'. You will need to detach and reattach tmux after doing this if you don't restart it.
If the inner tmux is very old you may also need to do set -g set-clipboard on for it as well.