Surface superconductors such as Si(111)-(root 7 x root 3)-In (referred to as (root 7 x root 3)-In, hereafter) can be sensitively doped by surface dopants to tune their transition temperature (T-c) even via a weak interaction induced by molecular overlayers. Here, we examined the effect of Cu-phthalocyanine (CuPc) and Cu-hexadecafluorophthalocyanine (F16CuPc) overlayers on the electronic structure on (root 7 x root 3)-In. While both molecules formed well-ordered monolayers, molecular adsorption geometries were found to be different; CuPc formed an ordered monolayer of face-on molecules, while F16CuPc tended to adsorb in a slightly tilted geometry. The normal-state electronic structures of (root 7 x root 3)-In covered with these molecules were found to be similar. Photoelectron spectroscy have suggested that the charge transfer to the molecules was negligible and that the interaction between the molecules and (root 7 x root 3)-In were weak. Nevertheless, CuPc enhanced the superconducting transition temperature (T-c) of (root 7 x root 3)-In by approximately 0.2 K, while F16CuPc suppressed it by approximately 1.0 K. The density functional theory (DFT) calculations combined with work function measurements showed that the surface electrons were accumulated in the region between the surface In atoms and the molecules, while molecules remain nearly uncharged. The increase in T-c in the case of CuPc overlayer is explained by a hole doping into (root 7 x root 3)-In due to the extraction of surface electrons. In contrast, the decrease in T-c for F16CuPc overlayer is attributable to the tilting adsorption geometry, possibly inducing an enhanced exchange interaction between spin magnetic moments of the molecular orbitals of inclined F16CuPc and the conduction electrons of (root 7 x root 3)-In. The high sensitivity of T-c to the small difference in the adsorption geometry of weakly interacting molecules suggests the possibilities of modifying the superconducting properties of various surface materials.