Inside tmux v2.4+
(relevant commit), you can define a binding to jump to the last prompts with acceptable reliability:
bind-key b copy-mode\;\
send-keys -X start-of-line\;\
send-keys -X search-backward " "
" " is a non-breaking space and a corresponding change is made to have you shell prompt contain it (
$PROMPT_COMMAND in bash).
It may require special care to set it in the shell prompt and the tmux configuration, as it could be changed to regular space during copy-paste operations (see method with vim in the comments).
Alternatively, you can try your favorite alternative unicode space character or anything that occurs rarely in command outputs and that you're ready to accept seeing in your prompt (🍌?).
This could also be a combination of several characters for even less false positives. It's probably better to pick something at the very end of the prompt, though, for easier navigation.