ComponentInRenderProps.js

class ContactDetails extends Component {
  render() {
    return (
      <SomeUtility>
        {(primitiveValue, SomeComponent) => (
          //QUESTION: you see that <SomeComponent /> component that <SomeUtility /> is providing in the
          //render props? Is it possible for some piece of state from <SomeUtility /> to be passed to
          //<SomeComponent /> ***automatically*** whenever it's rendered?

          //The best I can come up with is to *create* de novo <SomeComponent /> in <SomeUtility />'s constructor
          //and then pass *that*, but it seems hacky.

          //lots of cool additions to React lately - am I missing something?
          <div>
            {primitiveValue == 1 ? <SomeComponent val="a" /> : null}
            {primitiveValue == 2 ? <SomeComponent val="b" /> : null}
            {primitiveValue == 3 ? <SomeComponent val="c" /> : null}
          </div>
        )}
      </SomeUtility>
    );
  }
}

I usually use React.createElement(SomeComponent, props) which in your case would be parent's state that is passed as props. Don't know if this is what you need.
I created similar example where I wanted to to use uiWidget as component and then pass all parent's state to them, also various callbacks.