We numerically investigate the role of cladding geometries in two widely used anti-resonant hollow-core fiber designs with negative curvatures, the tubular negative-curvature fiber and ice-cream-cone negative-curvature fiber. The confinement loss governed by the inhibited coupling between the modes in the core and cladding is thoroughly examined systematically against the core-cladding curvature for both types. We show that, in addition to the mode-index mismatch, the mode-field overlap also plays a key role in determining the loss. Simultaneously, we find the ice-cream-cone negative-curvature fiber can exhibit better loss performance than the tubular design within a specific range of the curvature. This enhancement is achieved without sacrificing the transmission bandwidth and is relatively robust against the fabrication error.