Example.jsx

import { useState, useEffect } from "react";
import { useAuth } from "react-oidc-context";

const [UNAUTHENTICATED, AUTHENTICATING, AUTHENTICATED] = ["unauthenticated" , "authenticating", "authenticated"];

function Content(props) {
  switch (props.authState) {
    case UNAUTHENTICATED:
      return (
        <p>
          <button className="button" onClick={props.login}>
            Log in
          </button>
        </p>
      )
    case AUTHENTICATING:
      return (
        <p>Loading...</p>
      )
    case AUTHENTICATED:
      return (
        <>
          <h1 className="title">Hello, {props.user.name}!</h1>
          <p>
            Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do
            eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad
            minim veniam, quis nostrud exercitation ullamco laboris nisi ut
            aliquip ex ea commodo consequat. Duis aute irure dolor in
            reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla
            pariatur. Excepteur sint occaecat cupidatat non proident, sunt in
            culpa qui officia deserunt mollit anim id est laborum.
          </p>
          <p>
            <button className="button" onClick={props.logout}>
              Log out
            </button>
          </p>
        </>
      )
  }
}

export default function Example() {
  const auth = useAuth();
  const login = () => auth.signinRedirect();
  const logout = () => auth.removeUser();
  const [user, setUser] = useState(null);
  const authState =
    (!auth.isAuthenticated) ? UNAUTHENTICATED :
    (user == null) ? AUTHENTICATING :
    AUTHENTICATED;

  useEffect(() => {
    if (authState !== AUTHENTICATING) {
      return;
    }
    (async () => {
      try {
        const token = auth.user?.access_token;
        const response = await fetch("https://api.atlassian.com/me", {
          headers: {
            Authorization: `Bearer ${token}`,
          },
        });
        const user = await response.json();
        setUser(user);
      } catch (e) {
        console.error(e);
      }
    })();
  }, []);

  return (
    <section className="container">
      <Content authState={authState} user={user} login={login} logout={logout} />
    </section>
  )
}

動かないこともないけど、Exampleをマウントし直すとuserが消去されるし、そのたびfetchしているので、上位コンポーネントでfetchしたデータを引き回した方がいいのかも。(SWRを使うのは違うっぽい）