We consider the incompressible time-dependent Navier-Stokes system with Oseen term, in a 3D exterior domain, with the option of adding to the system another term arising in the study of stability of stationary incompressible Navier-Stokes flows. We do not impose any boundary conditions. The solutions we consider are supposed to possess properties of L^2-strong solutions: The velocity u is an L^∞-function in time and L^κ-integrable in space for some κ ∈ [1, 3) and some κ ∈ (3, ∞), the spatial gradient ∇_x u is L 2-integrable in space and in time, and the nonlinearity (u · ∇_x)u is L^2-integrable in time and L^3/2-integrable in space. We show that if the right-hand side of the equation and the initial data decay pointwise in space sufficiently fast, then u and ∇_x u also decay pointwise in space, with rates which are higher than those exhibited in previous articles.