Context. Coronal streamers appear to be strictly associated with the generation of the slow solar wind, even if a firm identification of the sources of the particle flux within these structures is still an unresolved issue. Aims. The purpose of this work is to contribute to a better knowledge of the physical characteristics of streamers and of their surroundings in a wide range of heliocentric distances and at both high radial and latitudinal resolutions. Methods. The analysis is based on spectral observations of a narrow, mid-latitude streamer performed with UVCS/SOHO during one week in May 2004: H Lyα and O VI resonance doublet line intensities and profiles were obtained at different heliocentric distances and latitudes. In addition, white-light polarized brightness images were taken in the same days of observation, through the LASCO/SOHO C2 coronagraph. Results. The radial variations in electron density and temperature, H I and O VI kinetic temperatures, and outflow velocities were derived from the observed line intensities, profiles, and O VI line intensity ratios between 1.6 and 5.0 R ⊙ , in two regions, 2-3 arcmin wide, located along the boundaries and in a narrow strip (5-10 arcmin) outside the streamer structure. Significantly high kinetic temperatures and outflow velocities were found in the out-of-streamer region above 3.0 R ⊙ for the Ovi ions and, for the first time, Hi atoms, compared to those obtained along the streamer boundaries. Moreover, the O VI kinetic temperatures and velocities turn out much higher than the H I ones at any heliocentric distance in all the observed regions. A higher anisotropy is also noticed for the O VI kinetic temperature in the region flanking the streamer. Conclusions. The slow coronal wind is found to flow with significantly different speeds and kinetic temperatures along the boundaries of the streamer and in the out-of-streamer regions at all heights, above 3.0-3.5 R ⊙ . This fact, consistent with previous studies, indicates that two components of slow wind probably form in the observed regions: one originates just above the streamer cusp and flows with velocities a little higher than 100 km s -1 , while the other flows along the open magnetic field lines flanking the streamer with velocities slightly lower than the slow wind asymptotic heliospheric value of ∼400 km s -1 , around 5.0 R ⊙ .