方法

useStateをモジュールに切り分ける。(下記appMode.tsの、function useAppMode参照)
一番外側のコンポーネントで、モジュール化されたuseState（下記のfunction useAppMode）を呼ぶ。
以後、任意のコンポーネントで、ステート（下記appMode.tsの mode）、ステートのsetter（下記appMode.tsのsetMode）が参照可能となる。

appMode.ts

import { useState, Dispatch, SetStateAction } from 'react'

export enum AppMode {
  home,
  write,
  read,
}

export let mode: AppMode
export let setMode: Dispatch<SetStateAction<AppMode>>

export function useAppMode() {
  ;[mode, setMode] = useState(AppMode.home)
}

例

appMode.ts
app.tsx
- home.tsx
- write.tsx
- read.tsx

app.tsx

import {
  isHomeMode,
  isWriteMode,
  isReadMode,
  useAppMode,
} from '../logic/appMode'
import { Reader } from '../components/reader'
import { Writer } from '../components/writer'
import { Home } from '../components/home'

export default () => {

  useAppMode() // モジュール化したuseStateを呼び出し

  return (
    <div id="app">
      {isHomeMode() ? <Home></Home> : null} // stateをgetterで参照
      {isWriteMode() ? <Writer></Writer> : null}
      {isReadMode() ? <Reader></Reader> : 
    </div>
  )
}

home.tsx

import { setWriteMode } from '../logic/appMode'
import { setReadMode } from '../logic/appMode'

export const Home = () => (
  <div>
    <button onClick={() => setWriteMode()}>Write</button>
    <button onClick={() => setReadMode()}>Read</button> // stateのsetterを呼び出し
  </div>
)

appMode.ts

import { useState, Dispatch, SetStateAction } from 'react' 

enum AppMode {
  home,
  write,
  read,
}

let mode: AppMode
let setMode: Dispatch<SetStateAction<AppMode>>

export function useAppMode() {
  ;[mode, setMode] = useState(AppMode.home)
}

// getter setter (※不要。上記、mode, setMode, AppModeをexportすればよい。可読性と保守性にメリット。)
export function isHomeMode(): boolean {
  return mode == AppMode.home
}
export function isWriteMode(): boolean {
  return mode == AppMode.write
}
export function isReadMode(): boolean {
  return mode == AppMode.read
}
export function setHomeMode(): void {
  setMode(AppMode.home)
}
export function setWriteMode(): void {
  setMode(AppMode.write)
}
export function setReadMode(): void {
  setMode(AppMode.read)
}