You can setup Putty to just log everything by adding logging to your Default Settings, that way whenever you connect, you’ll always be logging what the outputs are to your chosen location, e.g. your local machine. Although be aware of any sensitive on-screen information that would also be logged to these files.
Your path may vary, but the following path example will log the date and time along with the device name as the filename.
C:\puttylogs\&H-&T-&D-&M-&Y_putty.log