We report the spectroscopic analysis of 20 halo ab-type RR Lyrae stars with heliocentric distances between 15 and 165 kpc, conducted using medium-resolution spectra from the Magellan Inamori Kyocera Echelle (MIKE) spectrograph. We obtain the systemic line-of-sight velocities of our targets with typical uncertainties of 5-10 km s(-1) and compute orbital parameters for a subsample out to 50 kpc from the Galactic centre, including proper motion data from Gaia DR3. The orientation of our stars' orbits, determined for an isolated Milky Way and for a model perturbed by the Large Magellanic Cloud, appears to suggest an accreted origin for at least half of the sample. In addition, we derive atmospheric parameters and chemical abundance ratios for seven stars beyond 20 kpc. The derived alpha-abundances of five of these stars follow a Milky Way halo-like trend, while the other two display an underabundance of alpha-elements for their [Fe/H], indicating an association with accretion events. Furthermore, based on the [Sr/Ba] ratio, we can speculate about the conditions for the formation of a potential chemically peculiar carbon-enhanced metal-poor (CEMP) RR Lyrae star. By analysing the stars' orbital parameters and abundance ratios, we find hints of association of two of our stars with two massive satellites, namely the Large Magellanic Cloud and Sagittarius. Overall, our results are in line with the suggestion that the accretion of sub-haloes largely contributes to the outer halo stellar populations.