No, at least not for this example - if you have a function that is just awaiting a Promise and returning the result, you can just return that Promise alone, without any async or await:
function one() {
  return fetch(.....);
}
function two() {
  return one();
}
function three() {
  return two();
}
If you want to have a flat function body, you would need to use await when that function consumes a Promise and needs to do something else before returning another resolved Promise to the caller. For example:
async function one() {
  const fetchResult = await fetch(.....);
  // do something with fetchResult
  return foo;
}
function two() {
  return one();
}
function three() {
  return two();
}
Here, one is awaiting the fetch call, and doing something with it afterwards before returning a Promise, but two and three don't need to be async because, again, they're just calling a function that returns a Promise and returning that Promise to the caller. If two or three also had to do something after waiting but before resolving, then they would have to use await (if you wanted a flat function body):
async function one() {
  const fetchResult = await fetch(.....);
  // do something with fetchResult
  return foo;
}
async function two() {
  const oneResult = await one();
  console.log('got result for one');
  return oneResult;
}
function three() {
  return two();
}