Instead of just "ThreeMovers or another," you'll get better results by asking in local spaces:
Nextdoor or your local Facebook community group: "Looking for recommendations for a reliable mover from [Your City] to [New City]. Who did you have a great experience with?" This will yield hyper-local, recent referrals.
Reddit: Check your city's subreddit (e.g., r/AskNYC, r/Chicago) or r/moving for very candid advice.
My Final Advice
Start this process as early as possible. Good movers book weeks, sometimes months, in advance, especially during peak season (May-September). Once you narrow it down, trust your gut. If a company is difficult to communicate with during the estimate process, it will likely be worse on moving day.