The HDI you mentioned puts the USA in 3rd place, which is one of its better showings.
The one I had in mind when I posted was the Index of Freedom in the World from the Cato, Fraser, and Liberales Institutes, which puts the US in 7th place.
The Index of Economic Freedom, from the Heritage Foundation, puts the US in 10th place.
The World Press Freedom Index, from, Reporters Without Boarders, puts the US in 32nd place.
The Freedom Meta-Index at FreeExistence.Org is interesting because there is no possible metric in which the United States scores higher than New Zealand.
I have yet to see any study which concludes that the USA is the "most free" country in the world.