Compact higher-order (HO) schemes for a new finite difference method, referred to as the Cartesian cut-stencil FD method, for the numerical solution of the convection-diffusion equation in complex shaped domains have been addressed in this paper. The Cartesian cut-stencil FD method, which employs 1-D quadratic transformation functions to map a non-uniform (uncut or cut) physical stencil to a uniform computational stencil, can be combined with compact HO Padé-Hermitian formulations to produce HO cut-stencil schemes. The modified partial differential equation technique is used to develop formulas for the local truncation error for the cut-stencil HO formulations. The effect of various HO approximations for Neumann boundary conditions on the solution accuracy and global order of convergence are discussed. The numerical results for second-order and compact HO formulations of the Cartesian cut-stencil FD method have been compared for test problems using the method of manufactured solutions.