Photoproduction of a phi meson on protons was studied by means of linearly polarized photons at forward angles in the low-energy region from threshold to Egamma = 2.37 GeV. The differential cross sections at t = -|t|min do not increase smoothly as Egamma increases but show a local maximum at around 2.0 GeV. The angular distributions demonstrate that phi mesons are photoproduced predominantly by helicity-conserving processes, and the local maximum is not likely due to unnatural-parity processes.