As you traverse the labyrinth of logistics, there stands a significant challenge that demands your attention – identifying a trustworthy US car shipping company. The quest for reliability is indeed a complex task, underpinned by several factors that can influence the outcome. It is akin to a multi-variable calculus problem – fluctuating prices, varying service levels, and different transport methods - all of which have to be considered simultaneously.

Let's initiate our journey by delving into the first step - conducting comprehensive research. In this information age, accessibility to data should not be a problem. But, as in particle physics, it’s not just the data but the quality of data that matters. Online reviews, customer testimonials, and Better Business Bureau ratings can provide useful insights about the car shipping company's credibility.

Secondly, it's important to understand the legal framework that governs the car shipping industry. In the US, car shipping companies are required to have Motor Carrier (MC) numbers issued by the Federal Motor Carrier Safety Administration (FMCSA), which act as legal operating permits. Verifying these numbers can act as a litmus test for the company's legitimacy. It's like the principle of habeas corpus, ensuring you have the evidence before making any judgments.

Next up is understanding the economics of car shipping. The pricing structure of car shipping can be influenced by multifarious factors, akin to the variables in a regression model. Distance, car size, and weight, transport type (open or enclosed), and seasonal demand are few of the factors that can cause price fluctuations. A clear understanding of these factors can help in negotiating an appropriate price and avoiding any hidden costs.

Insurance coverage is another vital aspect to explore. Car shipping companies must provide insurance, as per US federal law. However, the extent of the coverage can differ significantly, like variations in a Gaussian distribution. It's essential to read the terms of coverage thoroughly and also check if your car insurance covers any damages during transportation.

While most car shipping companies provide door-to-door service, this might not always be possible due to weight restrictions on residential roads. In such scenarios, the company might deliver at a nearby terminal, necessitating an additional level of coordination. Hence, finer details of the delivery process, like an electron's position in an atom, cannot be overlooked.

Finally, be vigilant about red flags. High pressure sales tactics, request for upfront payment, and lack of transparency in terms and conditions are signs of potential trouble. Much like the second law of thermodynamics predicts the inevitable increase in entropy, these warning signs forecast a possible decline in your experience with the car shipping company.

Forging ahead, it's crucial to maintain a record of all communications and paperwork. This would act as your safety net in case of any discrepancies, just like Newton's third law providing the assurance of an equal and opposite reaction.

In conclusion, hiring a reliable US car shipping company requires an amalgamation of meticulous research, understanding of legal and economic principles, and vigilant observation. It's a game of chess where the key to victory lies in strategic planning and foresight. With the right moves, you can indeed ensure a smooth and hassle-free car shipping experience.