The thermocapillary flow in half-zone liquid bridges of low-Prandtl-number fluids, when the surface is partially confined by thin solid walls, is investigated numerically. The linear stabilities to three-dimensional disturbances and their physical destabilizing mechanisms are addressed for the axisymmetric steady flows, which are formed by changing the aspect ratio of the liquid bridge, the axial position of the free surface, and the ratio ξ of the free surface length to the full liquid height. A partial covering of the free surface adjacent to the cold disc significantly affects the axisymmetric steady toroidal vortices, especially the ellipticity of the streamlines. By decreasing ξ the effect of the elliptic mechanism on the energy production of the disturbance diminishes, leading to four distinct instability modes, including two oscillatory modes. Each of the instability modes essentially consists of the elliptic instability and / or the centrifugal instability. For a sufficiently small ξ , the instability appears to be purely centrifugal. The range of the ξ in which the oscillatory modes take place becomes narrower, the smaller the aspect ratio of the whole domain. If the free sueface is assigned on the cooling side, the instability is always the same as that for the unconfined half-zones.