I would say that the right to privacy or the expectation of it should be something seen as an inalienable right. The framers were able to construct a notion of freedom that allowed individuals to do go out and do great things, but also helped to create a realm where individuals have the right to be left alone. This aspect is not necessarily stated in the Constitution, but it is implied in such a strong way that it should be considered an inalienable right that can be only be intruded upon if an individual is posing a danger to themselves or to others. I think this is very much what Mill would be talking about in his conception of freedom. Outside of this, I think that there is an inalienable right to pursue one's right to property. This is something taken from John Locke and was the original basis for Jefferson's work, but I think that it is valid today. Individuals have to right to pursue their end of participating in a capitalist and free market venture of being able to generate income. Again, I think that when this infringes on the rights of another, there might have to be some intervention, yet I would consider this right to property and pursuit of economic rights, the right to make and keep money, an inalienable right.
When you say "additional" do you mean in addition to those set out in the United States's Declaration of Independence? If so, I would say that we do not. The reason for this is the three that are mentioned in the Declaration are so broad as to encompass any right you can imagine.
In the Declaration, we are said to have the rights to life, liberty, and the pursuit of happiness. The last two of these can include anything. For example, people have the right to marry more or less whoever they would like to (as long as they are opposite sex). This is a right that was not present in many US states when my parents got married in 1964. But the right to marry can easily be put under either or both of the last two categories of rights mentioned in the Declaration. In fact, I cannot think of any right that you can imagine that does not fall under one of those categories.