Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and can be used to measure the sizes of handles of Riemann surfaces of positive genus. We study some families of once-holed tori that are conformally embedded in target Riemann surfaces of conformal mappings of a given noncompact Riemann surface of genus one, and correct some results given in Masumoto (Math. Z. 257:453-464, 2007).