Is it possible to resolve async function without return keyword

Is it possible to resolve async function without return keyword

No.

There is no way to get a reference to the promise that the call to the async function created, but there really is no need to access that either (and btw, you cannot .resolve() a promise, you'd actually need to get access to the promise's resolving functions).

The whole point of async/await is to play nice with promises and other thenables. The idea is that every asynchronous function returns a promise, and that you don't have to promisify anything (but if you really have to, do it separately) - and in fact, $.get does return a (jQuery) promise. So simply write

async function doStuff() {
    //stuff...
    var result = await $.get('http://some/url/...');
    // stuff...
    return someValue;
}

If you really have a callback-taking function, use a simple

async function doStuff() {
    // stuff…
    return new Promise(function(resolve, reject) {
        $.get({
            url: 'http://some/url/...',
            success: resolve,
            error: reject
            // don't do other "stuff" in here
        });
    });
}

When you're returning a promise from a function there is no need for the function to be async, actually it's even bad because you are creating two promise!

function doStuff() {
    return new Promise(function(resolve, reject) {
        $.get('http://some/url/...', result => resolve(result));
    });
}

async function main() {
    const result = await doStuff();
}