The Leidenfrost effect, classically associated with drops levitating on their own vapor over hot solid surfaces, can also be observed over hot baths of nonvolatile liquids. In view of substrate fluidity, heat transfer through the bath to the drop should most certainly be dominated by convection and not by only conduction as in the solids, which may be instrumental for an efficient heat supply to the drop given typically poor thermal conductivity of the liquids. Here, we undertake an experimental and numerical study of the flow in a bath of silicone oil V20 induced by an overlying Leidenfrost drop, highlighting that a toroidal vortex is formed underneath the drop whose direction of circulation turns out to be different for drops of different liquids. We show that this is due to a shift in a delicate interplay between three mechanisms pulling in different directions: (i) shear stresses exerted by the vapor escaping from the gap between the bath and the drop, as well as (ii) buoyancy action and (iii) thermocapillary (Marangoni) stresses, both due to local evaporative cooling of the bath by the drop. Whatever the structure of this locally induced convection, its crucial heat transfer enhancing efficiency is readily confirmed in numerical simulations as favoring levitation.