In a companion paper, we investigated the question of the spatial origin of the cosmic rays detected in the Solar neighborhood, in the case of standard sources located in the Galactic disk. There are some reasons to believe that there may also be a large number of sources located in the halo, for example if the Galactic dark matter is made of supersymmetric particles or if Primordial Black Holes are present. These exotic sources could enhance the bar {p}, bar {d} or positrons above the standard background, indicating the existence of new physics. The spatial distribution of these hypothetical sources, though an important ingredient to evaluate these exotic signals, is poorly known. The aim of this paper is to point out that this discussion should not be disconnected from that of the propagation properties in the Galaxy. More precisely, we determine the regions of the halo from which a significant fraction f of cosmic rays antiprotons and antideuterons detected in the Solar neighborhood were emitted (we refer to these regions as f-volumes), for different sets of propagation parameters consistent with B/C data, as derived in \citet{Maurin02b}. It is found that some of them lead to rather small f-volumes, indicating that the exotic cosmic rays could have a local origin (in particular for a small diffusive halo or a large Galactic convective wind), coming from the Solar neighborhood or the Galactic center region. It is also found that the dark matter density enhancement (spike) due to the accretion around the central supermassive black hole gives a negligible contribution to the exotic charged particle signal on Earth. The case of electrons and positrons is also discussed.