Is it possible to let supervisord add timestamps to the outputs in
stderr_logfile? I couldn't find it anything about that in the documentation.
My $0.02. If you are running in Docker, then let the docker container capture stdin/stdout and capture timestamps that way. If not, send output to syslog and let either systemd or rsyslog append their timetstamps. Unless your system is heavily loaded (and rsyslog/systemd cannot keep up), this should get you what you need.
See this answer for docker.
See the official documentation for rsyslogd and search for "precision". For some vendors (RedHat, Ubuntu) this may already be enabled by default.
See this nice tutorial on using journalctl with systemd.