Some terminals (such as xterm) have a menu setting which enable/disables switching to/from alternate screen. Most do not...
By the way (same issue of some/most), if a terminal implements the original xterm 47 code alternate-screen switching, it is possible to switch back into the alternate screen without clearing it. I've noticed some which set TERM=xterm which do not implement this (only implementing the later 1049 code) - ymmv
You can see how the 47 code is used in a normal terminal description by looking at the smcup and rmcup capabilities from
infocmp xterm-old
which shows these items (among others):
rmcup=\E[2J\E[?47l\E8, smcup=\E7\E[?47h,
The \E[2J in rmcup clears the (alternate) screen before switching back to the normal screen. The \E7 and \E8 save/restore the cursor position (for the normal screen). If you really wanted to, you could replace those capabilities with whatever xterm-compatible terminal description, omitting the clear-screen part, allowing you to do
tput smcup tput rmcup
from the command-line and see what was on the alternate screen. It wouldn't be nice (since your command-line would overwrite parts of it), but it might help.
It's not needed in xterm, since xterm has a menu option for switching back/forth. But you might find that approach useful.
Further reading: