Possible extensions of the standard 3rd (upwind) and 4th (centered) order vertex-centered finite-volume advection schemes on unstructured triangular meshes are explored. A finite-volume analog of the P1 finite-element advection scheme with a consistent mass matrix is proposed, referred to as the compact scheme. In a simple two-dimensional test of advection by a shearing rotating flow, the compact scheme leads to a substantial error reduction compared with the traditionally used algorithms of the 3rd or 4th order, and can be nearly as accurate as the extension of the standard schemes to the 5th and 6th order. In full three-dimensional simulations of a turbulent baroclinically unstable flow, its eddy kinetic energy (EKE) only weakly increases for more accurate tracer advection. In terms of wall-clock CPU time, the compact scheme is the fastest despite the associated approximate inversion of mass matrix, and hence can be recommended.