Context. V-type asteroids in the inner main belt are thought to be genetically related to (4) Vesta as collisional fragments. We investigate their relationship with Vesta observing putative V-type asteroids. Aims. The aim of this work is to observe candidate V-type asteroids, selected in different regions of the inner main belt, to characterize them and hence better understand their relationship with (4) Vesta. Methods. We present new NIR reflectance spectra of 18 V-type candidate asteroids, selected from datasets of possible V-type asteroids. The data were obtained at the 3.6 m Telescopio Nazionale Galileo, covering the spectral range 0.7 to 2.5 mu m. We derive spectral parameters from NIR spectra to infer mineralogical information of the observed asteroids. The spectra of these asteroids are examined and compared to those of Howardite-Eucrite-Diogenite meteorites (HED), of which (4) Vesta is believed to be the parent body, and other V-type asteroids observed during previous campaigns. To enlarge the data set and increase the statistical significance of the analysis, we included the data presented in our previous article, obtaining a final data set of 41 V-type asteroids. Results. The V-type asteroids examined here show a higher variability of band parameters with respect to HEDs values, as for (4) Vesta. This result indicates that (4) Vesta and V-type asteroids have mineralogies that are not completely represented in the HEDs collection. Conclusions. An important finding is that some of the observed asteroids show spectral characteristics similar to diogenites, a result that may have relevant implications on their origin. Moreover, most of the diogenitic asteroids are not members of the Vesta classical family.