An electron crossing a vacuum medium interface generates transition radiation. The shape of this radiator can influence the angular distribution of the emission as well as impose a certain polarization. In this work we investigate whether transition radiation, induced inside a dielectric spiral phase plate (SPP) and registered by a detector centered along the electron propagation direction, can carry orbital angular momentum (OAM). The estimations for the UV transition radiation generated by an ultrarelativistic electron in both the wave zone and the intermediate zone reveal that the total angular momentum (TAM) vanishes in both regions, thus allowing one to obtain OAM using the spin angular momentum (SAM) estimates. It is further shown, that although the OAM of the entire radiation does not experience a significant change in between regions, its density distribution could undergo a shift from lower to higher polar angles in the intermediate zone. Thus, the apparent OAM seen by the two objects with similar finite angular size may differ relative to the radiator when one is placed in the wave zone and the other in the intermediate zone.