How can I refer to the value set by useState immediately afterwards?

You'd need to track it in a separate useEffect, where you receive the updated value:

useEffect(() => {
  console.log(campaigns);
}, [campaigns])

Another option is to use the value that was set on the state instead of the actual state value:

React.useEffect(() => {
  console.log('first use Effect');
  getRequest(`/api/v1/tablet/campaigns/) // getRequest return Promise Obj
  .then(result => {
    console.log(result); // [{...},{...},・・・,{...}]

    setCampaigns(result);

    console.log(result); // Access the result here, which is the same as campaigns
  });
}, []);

Actually it works you cant see campaigns in console because setCampaigns works async.

const [balance, setBalance] = React.useState(0);
const [campaigns, setCampaigns] = React.useState([]);

React.useEffect(() => {
  console.log('first use Effect');
  getRequest(`/api/v1/tablet/campaigns/) // getRequest return Promise Obj
  .then(result => {
    console.log(result); // [{...},{...},・・・,{...}]

    setCampaigns(result);//this function works

    console.log(campaigns); // but setCampaigns function works async
  });
}, []);

If you want to access it immediately you need to write below code that means when change campaigns object automatically trigger useeffect function

   React.useEffect(() => {         
        console.log(campaigns); // you can see your state data
      });
    }, [campaigns]);