Cone-beam computed tomography can reproduce a digital volume for an object that can be completely put inside the scan field of view (SFOV). In practice, the detector may not be wide enough to receive the projection, or the incident beam aperture may be confined to part of the object being scanning. Such cases cause width-truncated projections. With a dataset of width-truncated cone-beam projections, we can reconstruct a local volume in the object domain by using a convolution backprojection method. For the width-truncated projections, when the projection width is wider than the kernel length, we find that there exists, at the center of the object domain, a small region that can be tomographically reconstructed just as well as if there had been no truncation. Given the reconstruction kernel length, we propose a simple boundary extension technique to augment the SFOV, i.e., padding the truncated portions beyond the boundary. A continuous extrapolation beyond the boundary can effectively reduce the Gibbs effect, thereby enabling the reconstruction of a subvolume encompassing the SFOV. We demonstrate convolution-backprojection-based local cone-beam tomography by a breast phantom experiment, where the cone-beam projection is truncated by adjusting the x-ray collimator.