It wouldn't necessarily be a sadder note unless you had already been writing about a sad subject and this was even more sad. "Worse" means bad, not sad.
So: "The phone rang this morning, and it was my brother calling with sad news. His faithful hound dog Blue had been hit by a car and died. On an even sadder note, our aunt Wilma has just been diagnosed with cancer."
Or: "The phone rang this morning, and it was bad news: the contractor can't get started on the garage for at least another two weeks. On an even worse note, they're saying they don't expect the drought to end anytime soon."
Other alternatives: "on an unfortunate note," "on a less pleasant note." Or just, "I'm afraid I also have some sad/ bad news for you..."
[native speaker and writer, not a teacher]