Sometimes daemons do not terminate using the signal currently used by start-stop-daemon (sigterm?), therefore it would be nice if there was a flag that could be used with start-stop-daemon -K that would allow one to specify a signal.
If you are using the default start/stop functions, you can set the stopsig variable to the signal you want to send or set the retry variable to a retry schedule if you want to control the signal/wait times for stopping a daemon. If you are using start-stop-daemon directly, check out the -R and -s options. There is more info on the start-stop-daemon man page. Please let me know if those options cover what you need.
I was focused on two modes, start and stop, and forgot that s-s-d had a third mode, --signal, which accomplishes the goal.
