I believe i've figured this out. The key is to make multiple instances of the same shipping method, each one with it's own selection of shipping options.
So for example, I made 3 USPS shipping methods. One called first class mail. The 2nd called Priority mail. The 3rd called Express mail. Each with it's own set of restrictions, add-on-fees, notes, and relevant options selected.
Repeat for FedEx and UPS. And of course re-order in the "Shipping Methods" menu as you'd mentioned to set the default and relative order... mixing and matching FedEx, UPS, and USPS as you wish.
In hind sight, this seems so obvious but I couldn't find any tips on this anywhere, so I thought I'd document my learning for future users.
Hopefully this is the right way to accomplish this.