We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1-1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold V 2 R m+1 = SO(m + 1)/SO(m − 1) and all g-natural metrics on T 1 S m (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on V 2 R m+1 , m = 3, initially constructed by Kobayashi.