[en] The ability to accurately derive black hole (BH) masses at progressively higher redshifts and over a wide range of continuum luminosities has become indispensable in the era of large-area extragalactic spectroscopic surveys. In this paper, we present an extension of existing comparisons between rest-frame UV and optical virial BH mass estimators to intermediate redshifts and luminosities comparable to the local Hβ reverberation-mapped active galactic nuclei (AGNs). We focus on the Mg ii, C iv, and C iii] broad emission lines and compare them to both Hα and Hβ. We use newly acquired near-infrared spectra from the Fiber-fed Multi-object Spectrograph instrument on the Subaru telescope for 89 broad-lined AGNs at redshifts between 0.3 and 3.5, complemented by data from the AGES survey. We employ two different prescriptions for measuring the emission line widths and compare the results. We confirm that Mg ii shows a tight correlation with Hα and Hβ, with a scatter of ∼0.25 dex. The C iv and C iii] estimators, while showing larger scatter, are viable virial mass estimators after accounting for a trend with the UV-to-optical luminosity ratio. We find an intrinsic scatter of ∼0.37 dex between Balmer and carbon virial estimators by combining our data set with previous high redshift measurements. This updated comparison spans a total of three decades in BH mass. We calculate a virial factor for C iv/C iii] $\mathrm{l}\mathrm{o}\mathrm{g}{f}_{\mathrm{C}\mathrm{I}\mathrm{V}/\mathrm{C}\mathrm{I}\mathrm{I}\mathrm{I}]}=0.87$ with an estimated systematic uncertainty of ∼0.4 dex and find excellent agreement between the local reverberation mapped AGN sample and our high-z sample.