Considering the scenario example of the Average Total Costs wherein the example company had three machines and determined the amount of labor required to achieve optimal output, if the company were to use their increased productivity to fund an additional production machine, would they need to start from the beginning after purchase and operation with actual output data or could the model be adjusted to determine the necessary labor needed to produce the minimum average total cost?

I don't think that a company would have to start from the beginning to determine labor needed. I think that the model and formula could be adjusted or estimated to get a realistic answer.