The study is aimed at discovering new objects with the B[e] phenomenon in the Large Magellanic Cloud. We report medium-resolution optical spectroscopic observations of two newly found (ARDB54 and NOMAD0181-0125572) and two previously known (HenS-59 and HenS-137) supergiants with the B[e] phenomenon in the Large Magellanic Cloud. The observations were obtained with the GMOS spectrograph at the southern Gemini telescope. The optical spectra and the fundamental parameters of ARDB54 and NOMAD0181-0125572 are presented for the first time. We found that the Balmer line profiles of HenS-59 and HenS-137 were different from those observed in their spectra nearly 20 years ago. We suggest a higher effective temperature and luminosity for both objects. With the new fundamental parameters, the lowest luminosity for known supergiants with the B[e] phenomenon in the Magellanic Clouds is higher that previously thought (logL/L☉∼4.5 instead of 4.0). The object HenS-59 may be a binary system based on its UV excess, variable B-V color-index and radial velocity of emission lines, and periodically variable I-band brightness.