What I did in Alfred is this in the compose.yaml file:

  elasticsearch:
    [...]
    healthcheck:
      test:     "exit $( ( curl -s 'http://localhost:9200/_cat/indices' || echo red ) | grep '^red' | wc -l )"
      interval: '5s'
      retries:  10

And then docker compose up --wait.

Maybe that /_cat/health?h=st is better for this though. But the advantage of putting it in the compose means it works everywhere, rather than just for your upgrade make target.

Just FYI, in case it's useful.

I didn't bother with PostgreSQL or other services as they're pretty quick to start, but ES is (very) slow.