Your first sentence is not true. There is a whole family of freefall frames that are co-incident at any spacetime event. They correspond to different "initial velocities" as they diverge from that event: in geometric language, their origins follow the many different geodesics defined by different tangent vectors of the tangent space at that spacetime event, and the directions of their axes are those defined by Lie dragging along the flow of the different geodesics.
Or, put is in more concrete terms. Drop an $x-y-z$ frame out of a spaceship above the Earth: at the same time throw another one out to the side of the spaceship. They are both freefalling inertial frames, and, as long as they are near enough to one another that tidal terms in Earth's gravity are small over the space between them, a Lorentz boost defines the transformation between them, defined by their relative initial velocity.