It would be very useful to append something like this to every command I run interactively:
| tee /tmp/lastCommand to all commands.
A potential start might be to use ~/.inputrc to bind a key. (That's the configuration file for the GNU Readline library.) For example, using the letter, o, as a mnemonic for output ...
Control-o: " | tee /tmp/lastCommand"
One could bind to the Return/Enter key, but that action would have unintended effects.